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On isomorphisms of finite Cayley graphs—a survey

2002-09-01
Cai Heng Li

Analysing finite locally 𝑠-arc transitive graphs

2003-08-25
Michael Giudici , Cai Heng Li , Cheryl E. Praeger
We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and are either locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G,s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–arc transitive for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given a normal subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> preserves both local primitivity and local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–arc transitivity and leads us to study graphs where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.

Finite transitive permutation groups and finite vertex-transitive graphs

1997-01-01
Cheryl E. Praeger , Cai Heng Li , Alice C. Niemeyer

The Finite Primitive Permutation Groups Containing an Abelian Regular Subgroup

2003-10-23
Cai Heng Li
A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside … A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside in 1900. 2000 Mathematics Subject Classification 20B15, 20B30

Finite 2-arc-transitive abelian Cayley graphs

2006-12-31
Cai Heng Li , Jiangmin Pan

Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants

2005-03-01
Cai Heng Li
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On primitive permutation groups with small suborbits and their orbital graphs

2004-04-21
Cai Heng Li , Zai Ping Lu , Dragan Marušič

The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$

2001-04-24
Cai Heng Li
A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 … A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text {M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text {PSp}(4,p)$ with $p$ prime and $p\equiv \pm 1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.
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On edge-transitive Cayley graphs of valency four

2004-02-10
Xin Fang , Cai Heng Li , Ming Xu

On Half-Transitive Metacirculant Graphs of Prime-Power Order

2001-01-01
Cai Heng Li , Hyo-Seob Sim

Tetravalent edge-transitive Cayley graphs with odd number of vertices

2005-09-06
Cai Heng Li , Zai Ping Lu , Hua Zhang

On cubic Cayley graphs of finite simple groups

2002-02-01
Xin Fang , Cai Heng Li , Jie Wang +1 more

Finite edge-transitive Cayley graphs and rotary Cayley maps

2006-05-09
Cai Heng Li
This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley … This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

On partitioning the orbitals of a transitive permutation group

2002-09-19
Cai Heng Li , Cheryl E. Praeger
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>be a permutation group on a set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"><mml:semantics><mml:mi mathvariant="normal">Ω</mml:mi><mml:annotation encoding="application/x-tex">\Omega</mml:annotation></mml:semantics></mml:math></inline-formula>with a transitive normal subgroup<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"><mml:semantics><mml:mi>M</mml:mi><mml:annotation encoding="application/x-tex">M</mml:annotation></mml:semantics></mml:math></inline-formula>. Then<inline-formula content-type="math/mathml"><mml:math … Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>be a permutation group on a set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"><mml:semantics><mml:mi mathvariant="normal">Ω</mml:mi><mml:annotation encoding="application/x-tex">\Omega</mml:annotation></mml:semantics></mml:math></inline-formula>with a transitive normal subgroup<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"><mml:semantics><mml:mi>M</mml:mi><mml:annotation encoding="application/x-tex">M</mml:annotation></mml:semantics></mml:math></inline-formula>. Then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>acts on the set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper O normal r normal b normal l left-parenthesis upper M comma normal upper Omega right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mi mathvariant="normal">b</mml:mi><mml:mi mathvariant="normal">l</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Orbl}(M,\Omega )</mml:annotation></mml:semantics></mml:math></inline-formula>of nontrivial<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"><mml:semantics><mml:mi>M</mml:mi><mml:annotation encoding="application/x-tex">M</mml:annotation></mml:semantics></mml:math></inline-formula>-orbitals in the natural way, and here we are interested in the case where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper O normal r normal b normal l left-parenthesis upper M comma normal upper Omega right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mi mathvariant="normal">b</mml:mi><mml:mi mathvariant="normal">l</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Orbl}(M,\Omega )</mml:annotation></mml:semantics></mml:math></inline-formula>has a partition<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation></mml:semantics></mml:math></inline-formula>such that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>acts transitively on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation></mml:semantics></mml:math></inline-formula>. The problem of characterising such tuples<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma upper G comma normal upper Omega comma script upper P right-parenthesis"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">(M,G,\Omega ,\mathcal P)</mml:annotation></mml:semantics></mml:math></inline-formula>, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue script upper P EndAbsoluteValue"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">|\mathcal P|</mml:annotation></mml:semantics></mml:math></inline-formula>is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue script upper P EndAbsoluteValue equals 2"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">|\mathcal P|=2</mml:annotation></mml:semantics></mml:math></inline-formula>exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>-actions on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"><mml:semantics><mml:mi mathvariant="normal">Ω</mml:mi><mml:annotation encoding="application/x-tex">\Omega</mml:annotation></mml:semantics></mml:math></inline-formula>and on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation></mml:semantics></mml:math></inline-formula>, and gives some construction methods for TODs.

Finite <i>s</i> -Arc Transitive Graphs of Prime-Power Order

2001-03-01
Cai Heng Li
An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 ⩽ i ⩽ s and αi−1 ≠ αi+1 for 1 ⩽ i ⩽ … An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 ⩽ i ⩽ s and αi−1 ≠ αi+1 for 1 ⩽ i ⩽ s − 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s ⩾ 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K 2 m , 2 m , or s = 2 and Γ is a normal cover of one of the following 2-transitive graphs: K p m + 1 (the complete graph of order pm + 1), K 2 m , 2 m − 2 m K 2 (the complete bipartite graph of order 2m + 1 minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.)

On Finite s-Transitive Graphs of Odd Order

2001-03-01
Cai Heng Li

The finite primitive groups with soluble stabilizers, and the edge-primitive <i>s</i> -arc transitive graphs

2011-02-28
Cai Heng Li , Hua Zhang
Motivated by the study of several problems in algebraic graph theory, we study finite primitive permutation groups whose point stabilizers are soluble. Such primitive permutation groups are divided into three … Motivated by the study of several problems in algebraic graph theory, we study finite primitive permutation groups whose point stabilizers are soluble. Such primitive permutation groups are divided into three types: affine, almost simple and product action, and the product action type can be reduced to the almost simple type. This paper gives an explicit list of the soluble maximal subgroups of almost simple groups. The classification is then applied to classify edge-primitive s-arc transitive graphs with s ⩾ 4, solving a problem proposed by Richard M. Weiss (1999).

Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

2004-07-26
Cai Heng Li
In this paper, a characterisation is given of finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s … In this paper, a characterisation is given of finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s\ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, it is shown that, for any given integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k not-equals 7"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\not =7</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a finite set (maybe empty) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s element-of StartSet 3 comma 4 comma 5 comma 7 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">s\in \{3,4,5,7\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transitive Cayley graphs of valency <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are their normal covers. This indicates that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s\ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

A class of finite symmetric graphs with 2-arc transitive quotients

2000-07-01
Cai Heng Li , Cheryl E. Praeger , Sanming Zhou
Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition [Bscr ] with block size v. A framework for studying such graphs Γ was developed … Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition [Bscr ] with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph Γ[Bscr ] relative to [Bscr ], the bipartite subgraph Γ[B, C] of Γ induced by adjacent blocks B, C of Γ[Bscr ] and a certain 1-design [Dscr ](B) induced by a block B ∈ [Bscr ]. The present paper studies the case where the size k of the blocks of [Dscr ](B) satisfies k = v − 1. In the general case, where k = v − 1 [ges ] 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on [Bscr ]. We prove that [Dscr ](B) contains no repeated blocks if and only if Γ[Bscr ] is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as Γ[Bscr ] for some graph Γ with these properties. We prove further that Γ[B, C] ≅ Kv−1,v−1 if and only if Γ[Bscr ] is (G, 3)-arc transitive.

Arc-transitive graphs of square-free order and small valency

2016-07-01
Cai Heng Li , Zai Ping Lu , Wang Gai-xia

Locally <i>s</i>‐distance transitive graphs

2011-02-01
Alice Devillers , Michael Giudici , Cai Heng Li +1 more
Abstract We give a unified approach to analyzing, for each positive integer s , a class of finite connected graphs that contains all the distance transitive graphs as well as … Abstract We give a unified approach to analyzing, for each positive integer s , a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s ‐arc transitive graphs of diameter at least s . A graph is in the class if it is connected and if, for each vertex v , the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v , for each i from 1 to s . We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s ≥2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex‐orbits or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:176‐197, 2012
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Isomorphisms of finite Cayley graphs

1997-08-01
Cai Heng Li
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ … An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
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On the Weiss conjecture for finite locally primitive graphs

2000-02-01
Marston Conder , Cai Heng Li , Cheryl E. Praeger
Abstract A graph Γ is said to be locally primitive if, for each vertex α, the stabilizer in Aut Γ of α induces a primitive permutation group on the set … Abstract A graph Γ is said to be locally primitive if, for each vertex α, the stabilizer in Aut Γ of α induces a primitive permutation group on the set of vertices adjacent to α. In 1978, Richard Weiss conjectured that for a finite vertex-transitive locally primitive graph Γ , the number of automorphisms fixing a given vertex is bounded above by some function of the valency of Γ . In this paper we prove that the conjecture is true for finite non-bipartite graphsprovided that it is true in the case in which Aut Γ contains a locally primitive subgroup that is almost simple.
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Line graphs and geodesic transitivity

2012-06-01
Alice Devillers , Wei Jin , Cai Heng Li +1 more
For a graph Γ, a positive integer s and a subgroup G ≤ Aut(Γ), we prove that G is transitive on the set of s -arcs of Γ if and … For a graph Γ, a positive integer s and a subgroup G ≤ Aut(Γ), we prove that G is transitive on the set of s -arcs of Γ if and only if Γ has girth at least 2( s − 1) and G is transitive on the set of ( s − 1)-geodesics of its line graph. As applications, we first classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive. Secondly we prove that the only non-complete locally cyclic 2-geodesic transitive graphs are the octahedron and the icosahedron.
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Local 2-geodesic transitivity and clique graphs

2012-10-24
Alice Devillers , Wei Jin , Cai Heng Li +1 more

Finite locally-quasiprimitive graphs

2002-03-01
Cai Heng Li , Cheryl E. Praeger , Akshay Venkatesh +1 more

Homogeneous factorisations of complete graphs with edge-transitive factors

2008-02-25
Cai Heng Li , Tian Khoon Lim , Cheryl E. Praeger
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The finite simple groups with at most two fusion classes of every order

1996-01-01
Cai Heng Li , Cheryl E. Praeger
Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class … Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class of a ϵ G is the set {aσ | σ ϵ Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where: (i) G has at most two fusion classes of order i for every i (23 examples); and (ii) any two elements of G of the same order are fused or inversenfused. The examples in case (ii) are: A5, A6,L2(7),L2(8), L3(4), Sz(8), M11 and M23An application is given concerning isomorphisms of Cay ley graphs.

s-Arc-transitive graphs and normal subgroups

2014-09-17
Cai Heng Li , Ákos Seress , Shu Jiao Song

Homogeneous factorisations of graphs and digraphs

2004-09-17
Michael Giudici , Cai Heng Li , Primož Potočnik +1 more

On normal 2-geodesic transitive Cayley graphs

2013-09-11
Alice Devillers , Wei Jin , Cai Heng Li +1 more
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Finite 2‐Geodesic Transitive Graphs of Prime Valency

2014-11-07
Alice Devillers , Wei Jin , Cai Heng Li +1 more
Abstract We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that … Abstract We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3.
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SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS

2001-11-01
Cai Heng Li , Cheryl E. Praeger
A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such … A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

Enumeration and Representation Theory of Spin Space Groups

2024-08-28
Xiaobing Chen , Jun Ren , Yanzhou Zhu +7 more
Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. … Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic ordered materials. However, the basic theory of SSG has seldom been developed. In this work, we present a systematic study of the enumeration and the representation theory of the SSG. Starting from the 230 crystallographic space groups and finite translation groups with a maximum order of eight, we establish an extensive collection of over 100 000 SSGs under a four-index nomenclature as well as international notation. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. To facilitate the identification of the SSG, we develop an online program that can determine the SSG symmetries of any magnetic ordered crystal. Moreover, we derive the irreducible corepresentations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples, including a candidate altermagnet <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>RuO</a:mi></a:mrow><a:mn>2</a:mn></a:msub></a:mrow></a:math>, spiral spin polarization in the coplanar antiferromagnet <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>CeAuAl</c:mi></c:mrow><c:mrow><c:mn>3</c:mn></c:mrow></c:msub></c:mrow></c:math>, and geometric Hall effect in the noncoplanar antiferromagnet <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>CoNb</e:mi></e:mrow><e:mn>3</e:mn></e:msub><e:msub><e:mi mathvariant="normal">S</e:mi><e:mn>6</e:mn></e:msub></e:mrow></e:math>. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials. Published by the American Physical Society 2024
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Further restrictions on the structure of finite CI-groups

2007-01-30
Cai Heng Li , Zai Ping Lu , Péter P. Pálfy
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On finite edge-primitive and edge-quasiprimitive graphs

2009-09-17
Michael Giudici , Cai Heng Li
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Finite groups in which any two elements of the same order are either fused or inverse-fused

1997-01-01
Cai Heng Li , Cheryl E. Praeger
Elements a, b of a group G are said to be fused or inverse-fused if there exists σεAut(G) such that a = bσ or a = (b-1)σ respectively. This paper … Elements a, b of a group G are said to be fused or inverse-fused if there exists σεAut(G) such that a = bσ or a = (b-1)σ respectively. This paper gives a classification of all finite groups in which any two elements of the same order are fused orinverse-fused.

LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

2009-02-01
Cai Heng Li , Jiangmin Pan , Li Ma
Abstract Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, … Abstract Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ × l , where Σ= K p m with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.
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On finite permutation groups with a transitive cyclic subgroup

2011-10-31
Cai Heng Li , Cheryl E. Praeger

Finite CI-Groups are Soluble

1999-07-01
Cai Heng Li
For a finite group G and a subset S of G with 1 ∉ S and S = S−1, the Cayley graph Cay(G, S) is the graph with vertex set … For a finite group G and a subset S of G with 1 ∉ S and S = S−1, the Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx−1 ∈ S. The group G is called a CI-group if, for all subsets S and T of G∖{1}, Cay(G, S) ≅ Cay(G, T) if and only if Sσ = T for some σ ∈ Aut(G). In this paper, for each prime p ≡ 1 (mod 4), a symmetric graph Γ(p) is constructed from PSL(2, p) such that Aut Γ(p) = Z2 × PSL(2, p); it is then shown that A5 is not a CI-group, and that all finite CI-groups are soluble. 1991 Mathematics Subject Classification 05C25.

Factorizations of Almost Simple Groups with a Solvable Factor, and Cayley Graphs of Solvable Groups

2022-07-26
Cai Heng Li , Binzhou Xia
A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and … A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied to study <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary that, except for cycles, a non-bipartite connected <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc-transitive Cayley graph of a finite solvable group is necessarily a normal cover of the Petersen graph or the Hoffman-Singleton graph.
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The locally 2-arc transitive graphs admitting a Ree simple group

2004-11-09
Xin Fang , Cai Heng Li , Cheryl E. Praeger

Characterizing finite locally s-arc transitive graphs with a star normal quotient

2006-01-01
Michael Giudici , Cai Heng Li , Cheryl E. Praeger
Let Γ be a finite locally (G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of … Let Γ be a finite locally (G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G-orbits. In previous work the authors showed that if G has a non-trivial normal subgroup intransitive on both of the vertex orbits of G, then Γ is a cover of a smaller locally s-arc transitive graph. Thus the ‘basic’ graphs to study are those for which G acts quasiprimitively on at least one of the two orbits. In this paper we investigate the case where G is quasiprimitive on only one of the two G-orbits. Such graphs have a normal quotient which is a star. We construct several infinite families of locally 3-arc transitive graphs and prove characterization results for several of the possible quasiprimitive types for G.

Regular maps whose groups do not act faithfully on vertices, edges, or faces

2004-01-27
Cai Heng Li , Jozef Širáň

On orbital regular graphs and frobenius graphs

1998-03-01
Xin Fang , Cai Heng Li , Cheryl E. Praeger

On isomorphisms of connected Cayley graphs, III

1998-08-01
Cai Heng Li
For a finite group G and a subset S of G which does not contain the identity of G , we use Cay( G, S ) to denote the Cayley … For a finite group G and a subset S of G which does not contain the identity of G , we use Cay( G, S ) to denote the Cayley graph of G with respect to S . For a positive integer m , the group G is called a (connected) m -DCI-group if for any (connected) Cayley graphs Cay( G, S ) and Cay( G, T ) of out-valency at most m , S σ = T for some σ ∈ Aut( G ) whenever Cay( G, S ) ≅ Cay( G, T ). Let p ( G ) be the smallest prime divisor of | G |. It was previously shown that each finite group G is a connected m -DCI-group for m ≤ p ( G ) − 1 but this is not necessarily true for m = p ( G ). This leads to a natural question: which groups G are connected p ( G )-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q ). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A 5 .
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A characterization of metacirculants

2012-07-13
Cai Heng Li , Shu Jiao Song , Dian Jun Wang

On self-complementary vertex-transitive graphs

1997-01-01
Cai Heng Li
In this paper we study the problem of determining positive integers n for which there exist self-complementary vertex-transitive graphs of order n. The main aim of this paper is to … In this paper we study the problem of determining positive integers n for which there exist self-complementary vertex-transitive graphs of order n. The main aim of this paper is to prove that, for distinct primes p, q, there exist self-complementary vertex-transitive graphs of order pq if and only if p, q Scz.tbnd;1 (mod 4). This provides negative answers to a long-standing open question proposed by Zelinka (1979) and a recent open question proposed by Froncek, Rosa Siran (1996).

Constructions of Self-complementary Circulants with No Multiplicative Isomorphisms

2001-11-01
Robert Jajcay , Cai Heng Li

The finite edge-primitive pentavalent graphs

2012-11-29
Songtao Guo , Yan‐Quan Feng , Cai Heng Li

The Exceptional Hall Numbers

2025-05-30
Zaibing Guo , Yong Hu , Cai Heng Li

The finite groups with three automorphism orbits

2025-05-01
Cai Heng Li , Yan Zhou Zhu

Arc-transitive maps with edge number coprime to the Euler characteristic -- I

2024-12-24
Cai Heng Li , Longxiang Liu
This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one establishes … This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one establishes a framework and carries out the classification work for arc-transitive maps with solvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the constructions of various new and interesting arc-regular maps.
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Regular maps with square-free Euler characteristic

2024-12-19
Pei Hua , Cai Heng Li , J. Zhang +1 more
In this paper, we give a characterization of finite non-solvable groups which act flag-regularly on maps with negative square-free Euler characteristic. We also construct some interesting families of regular maps. In this paper, we give a characterization of finite non-solvable groups which act flag-regularly on maps with negative square-free Euler characteristic. We also construct some interesting families of regular maps.

Finite semiprimitive permutation groups of rank $3$

2024-12-04
Cai Heng Li , Hanyue Yi , Yan Zhou Zhu
A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups … A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately transitive groups.The latter three classes of groups of rank $3$ have been classified, forming significant progresses on the long-standing problem of classifying permutation groups of rank $3$.In this paper, a complete classification is given of finite semiprimitive groups of rank $3$ that are not innately transitive, examples of which are certain Schur coverings of certain almost simple $2$-transitive groups, and three exceptional small groups.
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Finite imprimitive rank $3$ affine groups -- I

2024-12-03
Cai Heng Li , Hanyue Yi , Yan Zhou Zhu
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such … This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.
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On finite permutation groups of rank three

2024-12-01
Hong Yi Huang , Cai Heng Li , Yan Zhou Zhu
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A proof of gross' conjecture on 2‐automorphic 2‐groups

2024-10-25
Cai Heng Li , Yan Zhou Zhu
Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on … Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on 2‐automorphic 2‐groups, which are 2‐groups with involutions forming a single fusion class. As a consequence, two open problems regarding AT‐groups and FIF‐groups are solved.
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Orientable Vertex Primitive Complete Maps

2024-10-07
Xue Yu , Cai Heng Li , Ben Gong Lou

Coverings of Groups, Regular Dessins, and Surfaces

2024-09-03
Jiyong Chen , Wenwen Fan , Cai Heng Li +1 more
A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins … A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins as coverings of unicellular regular dessins. It shows that there are exactly three O'Nan-Scott-Praeger types of face-quasiprimitive regular dessins which are smooth coverings of unicellular regular dessins, leading to new constructions of interesting families of regular dessins. Finally, a problem of determining smooth Schur covering of simple groups is initiated by studying coverings between $\SL(2,p)$ and $\PSL(2,p)$, giving rise to interesting regular dessins like Fibonacci coverings.
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Enumeration and Representation Theory of Spin Space Groups

2024-08-28
Xiaobing Chen , Jun Ren , Yanzhou Zhu +7 more
Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. … Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic ordered materials. However, the basic theory of SSG has seldom been developed. In this work, we present a systematic study of the enumeration and the representation theory of the SSG. Starting from the 230 crystallographic space groups and finite translation groups with a maximum order of eight, we establish an extensive collection of over 100 000 SSGs under a four-index nomenclature as well as international notation. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. To facilitate the identification of the SSG, we develop an online program that can determine the SSG symmetries of any magnetic ordered crystal. Moreover, we derive the irreducible corepresentations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples, including a candidate altermagnet <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>RuO</a:mi></a:mrow><a:mn>2</a:mn></a:msub></a:mrow></a:math>, spiral spin polarization in the coplanar antiferromagnet <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>CeAuAl</c:mi></c:mrow><c:mrow><c:mn>3</c:mn></c:mrow></c:msub></c:mrow></c:math>, and geometric Hall effect in the noncoplanar antiferromagnet <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>CoNb</e:mi></e:mrow><e:mn>3</e:mn></e:msub><e:msub><e:mi mathvariant="normal">S</e:mi><e:mn>6</e:mn></e:msub></e:mrow></e:math>. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials. Published by the American Physical Society 2024
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Unipotent radicals, one-dimensional transitive groups, and solvable factors of classical groups

2024-07-17
Tao Feng , Cai Heng Li , Conghui Li +3 more
By developing a tangible way to decompose unipotent radicals into irreducible submodules of Singer cycles, we achieve a classification of solvable factors of finite classical groups of Lie type. This … By developing a tangible way to decompose unipotent radicals into irreducible submodules of Singer cycles, we achieve a classification of solvable factors of finite classical groups of Lie type. This completes previous work on factorizations of classical groups with a solvable factor. In particular, it resolves the final uncertain case in the long-standing problem of determining exact factorizations of almost simple groups. As a byproduct of the classification, we also obtain a new characterization of one-dimensional transitive groups, offering further insights into their group structure.
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Locally finite vertex-rotary maps and coset graphs with finite valency and finite edge multiplicity

2024-06-11
Cai Heng Li , Cheryl E. Praeger , Shu Jiao Song
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Covers and pseudocovers of symmetric graphs

2024-05-06
Cai Heng Li , Yan Zhou Zhu

On the EKR-module property

2024-04-29
Cai Heng Li , Venkata Raghu Tej Pantangi
In recent years, the generalization of the Erdős–Ko–Rado (EKR) theorem to permutation groups has been of much interest. A transitive group is said to satisfy the EKR-module property if the … In recent years, the generalization of the Erdős–Ko–Rado (EKR) theorem to permutation groups has been of much interest. A transitive group is said to satisfy the EKR-module property if the characteristic vector of every maximum intersecting set is a linear combination of the characteristic vectors of cosets of stabilizers of points. This generalization of the well-known permutation group version of the Erdős–Ko–Rado (EKR) theorem was introduced by K. Meagher in [28]. In this article, we present several infinite families of permutation groups satisfying the EKR-module property, which shows that permutation groups satisfying this property are quite diverse.
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Fixers and derangements of finite permutation groups

2024-04-29
Hong Yi Huang , Cai Heng Li , Yi Lin Xie
Let $G\leqslant\mathrm{Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. We say that a subgroup $K$ of $G$ is a fixer if every element of $K$ has fixed … Let $G\leqslant\mathrm{Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. We say that a subgroup $K$ of $G$ is a fixer if every element of $K$ has fixed points, and we say that $K$ is large if $|K| \geqslant |H|$. There is a special interest in studying large fixers due to connections with Erd\H{o}s-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle $\mathrm{PSL}_2(q)$, and we use this result to verify a special case of a conjecture of Spiga on permutation characters. We also present some results on large fixers of almost simple primitive groups with socle an alternating or sporadic group.
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Arc-transitive maps with coprime Euler characteristic and edge number

2024-03-27
Cai Heng Li , Lu Yi Liu
This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries … This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries out the classification work for arc-transitive maps with nonsolvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the construction of some new and interesting arc-regular maps.
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Intersecting subsets in finite permutation groups

2024-03-26
Cai Heng Li , Venkata Raghu Tej Pantangi , Shu Jiao Song +1 more
A subset (subgroup) $S$ of a transitive permutation group $G\leq Sym(\Omega)$ is called an intersecting subset (subgroup, resp.) if the ratio $xy^{-1}$ of any elements $x,y\in S$ fixes some point. … A subset (subgroup) $S$ of a transitive permutation group $G\leq Sym(\Omega)$ is called an intersecting subset (subgroup, resp.) if the ratio $xy^{-1}$ of any elements $x,y\in S$ fixes some point. A transitive group is said to have the EKR property if the size of each intersecting subset is at most the order of the point stabilizer. A nice result of Meagher-Spiga-Tiep (2016) says that 2-transitive permutation groups have the EKR property. In this paper, we systematically study intersecting subsets in more general transitive permutation groups, including primitive (quasiprimitive) groups, rank-3 groups, Suzuki groups, and some special solvable groups. We present new families of groups that have the EKR property, and various families of groups that do not have the EKR property. This paper significantly improves the unpublished version of this paper, and particularly solves Problem 1.4 of it.
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Locally-primitive block designs

2024-03-25
Jianfu Chen , Peice Hua , Cai Heng Li +1 more
A locally-primitive design is a block design $(\mathcal{P},\mathcal{B})$ which admits an automorphism group $G$ with primitive local actions. It is proved that $G$ is primitive on the points $\mathcal{P}$, and … A locally-primitive design is a block design $(\mathcal{P},\mathcal{B})$ which admits an automorphism group $G$ with primitive local actions. It is proved that $G$ is primitive on the points $\mathcal{P}$, and either $G$ is an almost simple group, or $G$ acting on $\mathcal{P}$ is an affine group.
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The finite groups with three automorphism orbits

2024-03-03
Cai Heng Li , Yan Zhou Zhu
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G.\,Higman in 1963. As … A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G.\,Higman in 1963. As a consequence, a classification is obtained for finite permutation groups of rank $3$ which are holomorphs of groups.
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The Factorizations of Finite Classical Groups

2024-02-28
Cai Heng Li , Lei Wang , Binzhou Xia
We classify the factorizations of finite classical groups with nonsolvable factors, completing the classification of factorizations of finite almost simple groups. We classify the factorizations of finite classical groups with nonsolvable factors, completing the classification of factorizations of finite almost simple groups.
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The exact factorizations of almost simple groups

2023-07-07
Cai Heng Li , Lei Wang , Binzhou Xia
Abstract This paper presents a classification of exact factorizations of almost simple groups, which has been a long‐standing open problem initiated around 1980 by the work of Wiegold–Williamson, and significantly … Abstract This paper presents a classification of exact factorizations of almost simple groups, which has been a long‐standing open problem initiated around 1980 by the work of Wiegold–Williamson, and significantly progressed by Liebeck, Praeger, and Saxl in 2010. The classification is then used to solve problems in bicrossproduct Hopf algebras and permutation groups.

Two-arc-transitive graphs of odd order: I

2023-04-05
Cai Heng Li , Jing Jian Li , Zai Ping Lu

Enumeration and representation of spin space groups

2023-01-01
Jun Ren , Xiaobing Chen , Yanzhou Zhu +5 more
Those fundamental properties, such as phase transitions, Weyl fermions and spin excitation, in all magnetic ordered materials was ultimately believed to rely on the symmetry theory of magnetic space groups. … Those fundamental properties, such as phase transitions, Weyl fermions and spin excitation, in all magnetic ordered materials was ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and physical properties of magnetic ordered materials such as altermagnets. However, the basic theory of SSG has been seldomly developed. In this work, we present a systematic study of the enumeration and the representation theory of SSG. Starting from the 230 crystallographic space groups and finite translational groups with a maximum order of 8, we establish an extensive collection of over 80,000 SSGs under a four-segment nomenclature. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. Moreover, we derive the irreducible co-representations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSGs and band degeneracies resulting from SSG symmetries through several representative material examples, including a well-known altermagnet RuO2, and a spiral magnet CeAuAl3. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials.

Finite permutation groups of rank $3$

2023-01-01
Hong Huang , Cai Heng Li , Yan Zhou Zhu
The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general … The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$.
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A Proof of Gross' Conjecture on $2$-Automorphic $2$-Groups

2023-01-01
Cai Heng Li , Yan Zhou Zhu
Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 … Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 on the so-called Suzuki $2$-groups, and confirms a conjecture of Gross proposed in 1974. As a consequence, two open problems on AT-groups and FIF-groups are solved.
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Complete circular regular dessins of coprime orders

2022-09-28
Wenwen Fan , Cai Heng Li , Shouhong Qıao

Factorizations of Almost Simple Groups with a Solvable Factor, and Cayley Graphs of Solvable Groups

2022-07-26
Cai Heng Li , Binzhou Xia
A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and … A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied to study <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary that, except for cycles, a non-bipartite connected <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc-transitive Cayley graph of a finite solvable group is necessarily a normal cover of the Petersen graph or the Hoffman-Singleton graph.
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The graphs with a symmetrical Euler cycle

2022-02-15
Jiyong Chen , Cai Heng Li , Cheryl E. Praeger +1 more
The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, … The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is a sequence (e1,e2,…,eℓ) of these ℓ edges, each occurring exactly once, such that ei, ei + 1 are incident with a common vertex for each i (reading subscripts modulo ℓ). An Euler cycle is symmetrical if there exists an automorphism of the graph such that ei → ei + 2 for each i. The cyclic group generated by this automorphism has one orbit on edges if ℓ is odd, or two orbits of length ℓ/2 if ℓ is even: that is to say, the group is regular or bi-regular on edges, respectively. Symmetrical Euler cycles arise naturally from arc-transitive embeddings of graphs in surfaces since, for each face of the embedded graph, the sequence of edges on the boundary of the face forms a symmetrical Euler cycle for the induced subgraph on this edge-set. We first classify all finite connected graphs which admit a cyclic subgroup of automorphisms that is regular or bi-regular on edges, and identify more than a dozen infinite families of examples. We then prove that exactly six of these families consist of graphs with symmetrical Euler cycles. These are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.
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Vertex-quasiprimitive locally primitive Graphs

2022-01-09
Ben Gong Lou , Cai Heng Li

Locally Finite Vertex-Rotary Maps and Coset Graphs with Finite Valency and Finite Edge Multiplicity

2022-01-01
Cai Heng Li , Cheryl E. Praeger , Shu Jiao Song
It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs … It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite edge-multiplicity, where $H, J$ are stabilisers in $G$ of a vertex and incident edge, respectively. Given a group $G=\l a,z\r$ with $|z|=2$ and $|a|$ finite, the coset graph $\Cos(G,\l a\r,\l z\r)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a $G$-arc-transitive map $(V,E,F)$, namely, a {\it $G$-rotary} map if $|az|$ is finite, and a {\it $G$-bi-rotary} map if $|zz^a|$ is finite. The $G$-rotary map can be represented as a coset geometry for $G$, extending the notion of a coset graph. However the $G$-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. We also give a coset geometry construction of a flag-regular map $(V,E,F)$. In all of these constructions we prove that the face boundary cycles are regular cycles which are simple cycles precisely when the given group acts faithfully on $V\cup F$.
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On the EKR Module property

2022-01-01
Cai Heng Li , Venkata Raghu Tej
We investigate permutation group satisfying the \emph{EKR-Module property}. This property gives a characterization of the maximum intersecting sets of permutations in the group. Specifically, the characteristic vector of a maximum … We investigate permutation group satisfying the \emph{EKR-Module property}. This property gives a characterization of the maximum intersecting sets of permutations in the group. Specifically, the characteristic vector of a maximum intersecting set is a linear combination of the characteristic vectors of cosets of stabilizer subgroups. Recently Meagher and Sin showed that all $2$-transitive groups satisfy the EKR-Module property. In this article we find a few more infinite classes of permutation groups satisfying this property.
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Factorizations of almost simple orthogonal groups of plus type

2022-01-01
Cai Heng Li , Lei Wang , Binzhou Xia
This is the fifth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of plus … This is the fifth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of plus type.
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Covers and Pseudocovers of Symmetric Graphs

2022-01-01
Cai Heng Li , Yan Zhou Zhu
We introduce the concept of {\it pseudocover}, which is a counterpart of {\it cover}, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given … We introduce the concept of {\it pseudocover}, which is a counterpart of {\it cover}, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger-Xu's graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by $32$ has connected pseudocovers.
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Skew-morphisms of nonabelian characteristically simple groups

2021-10-01
Jiyong Chen , Shaofei Du , Cai Heng Li
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Finite quasiprimitive permutation groups with a metacyclic transitive subgroup

2021-09-06
Cai Heng Li , Jiangmin Pan , Binzhou Xia
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Systems of imprimitivity for wreath products

2021-08-19
Mikko Korhonen , Cai Heng Li
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Two-arc-transitive graphs of odd order -- II

2021-05-09
Cai Heng Li , Jing Jian Li , Zai Ping Lu
It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then … It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an alternating group or a symmetric group. This is the second of a series of papers aiming towards a classification of 2-arc-transitive graphs of odd order.

Two-arc-transitive graphs of odd order — II

2021-05-08
Cai Heng Li , Jing Jian Li , Zai Ping Lu
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Systems of imprimitivity for wreath products

2021-05-07
Mikko Korhonen , Cai Heng Li
Let $G$ be an irreducible imprimitive subgroup of $\operatorname{GL}_n(\mathbb{F})$, where $\mathbb{F}$ is a field. Any system of imprimitivity for $G$ can be refined to a nonrefinable system of imprimitivity, and … Let $G$ be an irreducible imprimitive subgroup of $\operatorname{GL}_n(\mathbb{F})$, where $\mathbb{F}$ is a field. Any system of imprimitivity for $G$ can be refined to a nonrefinable system of imprimitivity, and we consider the question of when such a refinement is unique. Examples show that $G$ can have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where $G$ is the wreath product of an irreducible primitive $H \leq \operatorname{GL}_d(\mathbb{F})$ and transitive $K \leq S_k$, where $n = dk$. We show that $G$ has a unique nonrefinable system of imprimitivity, except in the following special case: $d = 1$, $n = k$ is even, $|H| = 2$, and $K$ is a transitive subgroup of $C_2 \wr S_{n/2}$. As a simple application, we prove results about inclusions between wreath product subgroups.

An explicit characterization of arc-transitive circulants

2021-03-05
Cai Heng Li , Binzhou Xia , Sanming Zhou
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NEW CONSTRUCTIONS OF SELF-COMPLEMENTARY CAYLEY GRAPHS

2021-01-12
Cai Heng Li , G.V.V. Jagannadha Rao , Shu Jiao Song
Abstract Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc. … Abstract Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc. 356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

Factorizations of almost simple linear groups

2021-01-01
Cai Heng Li , Lei Wang , Binzhou Xia
This is the first one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple linear groups. This is the first one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple linear groups.

Factorizations of almost simple unitary groups

2021-01-01
Cai Heng Li , Lei Wang , Binzhou Xia
This is the second one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple unitary groups. This is the second one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple unitary groups.

Factorizations of almost simple orthogonal groups in odd dimension

2021-01-01
Cai Heng Li , Lei Wang , Binzhou Xia
This is the third one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups in odd … This is the third one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups in odd dimension.

Factorizations of almost simple orthogonal groups of minus type

2021-01-01
Cai Heng Li , Wang Lei , Binzhou Xia
This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus … This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus type.
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The graphs with a symmetrical Euler cycle

2021-01-01
Jiyong Chen , Cai Heng Li , Cheryl E. Praeger +1 more
The edges surrounding a face of a map $M$ form a cycle $C$, called the boundary cycle of the face, and $C$ is often not a simple cycle. If the … The edges surrounding a face of a map $M$ form a cycle $C$, called the boundary cycle of the face, and $C$ is often not a simple cycle. If the map $M$ is arc-transitive, then there is a cyclic subgroup of automorphisms of $M$ which leaves $C$ invariant and is bi-regular on the edges of the induced subgraph $[C]$; that is to say, $C$ is a symmetrical Euler cycle of $[C]$. In this paper we determine the family of graphs (which may have multiple edges) whose edge-sets can be sequenced to form a symmetrical Euler cycle. We first classify all graphs which have a cyclic subgroup of automorphisms acting bi-regularly on edges. We then apply this classification to obtain the graphs possessing a symmetrical Euler cycle, and therefore are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.
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Two-arc-transitive graphs of odd order -- II

2021-01-01
Cai Heng Li , Jing Jian Li , Zai Ping Lu
It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then … It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an alternating group or a symmetric group. This is the second of a series of papers aiming towards a classification of 2-arc-transitive graphs of odd order.

Regular subgroups of quasiprimitive permutation groups.

2020-12-17
Cai Heng Li , Lei Wang , Binzhou Xia
Characterizing permutation groups $G$ containing a regular subgroup $H$ is a classical problem in permutation group theory, dated back to Burnside in the 19th century when he proved that a … Characterizing permutation groups $G$ containing a regular subgroup $H$ is a classical problem in permutation group theory, dated back to Burnside in the 19th century when he proved that a primitive group containing a regular cyclic subgroup of prime-power order is 2-transitive or of prime degree. Studying this problem has played a significant role in the history of permutation group theory. The problem has been solved for various important special cases, and among them a significant result achieved by Liebeck, Praeger and Saxl in 2010 solves the problem for the case where $G$ is primitive and almost simple. Praeger developed a theory for quasiprimitive permutation groups in 1992, which extended research scope of permutation group theory. Many problems in the applications of permutation groups can be reduced to the quasiprimitive case instead of the primitive case, and thus quasiprimitive permutation groups have been extensively studied during the past 30 years. In this paper, we classify the pairs $H<G$ for two cases, namely, $G$ is almost simple, or $G$ is quasiprimitive and $H$ is simple, which extend the work by Liebeck, Praeger and Saxl in 2010. One would expect that these results will have many applications. In particular, we confirm a conjecture of Etingof, Gelaki, Guralnick and Saxl in 2000 on bicrossproduct Hopf algebras, and show that $3$-arc transitive Cayley graphs of simple groups are surprisingly rare.

On solvable factors of almost simple groups

2020-12-04
Timothy C. Burness , Cai Heng Li
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Coauthor Papers Together
Cheryl E. Praeger 67
Michael Giudici 50
Alice Devillers 25
Binzhou Xia 23
Shu Jiao Song 17
Zai Ping Lu 13
Ákos Seress 12
Jiangmin Pan 11
Yan Zhou Zhu 11
Wei Jin 10
Gabriel Verret 9
Lei Wang 8
Wenwen Fan 7
Sanming Zhou 6
Primož Potočnik 6
John Bamberg 6
Jiyong Chen 6
Hua Zhang 6
Xin Fang 6
Eric Swartz 6
Ming Xu 5
Geoffrey Pearce 4
S. P. Glasby 4
Venkata Raghu Tej Pantangi 4
Marston Conder 4
Jing Jian Li 3
Yan‐Quan Feng 3
Shouhong Qıao 3
G.V.V. Jagannadha Rao 3
Hyo-Seob Sim 3
Jozef Širáň 3
Gai Xia Wang 3
Timothy C. Burness 3
V. I. Trofimov 3
Jin‐Xin Zhou 2
Jiayu Li 2
Yian Xu 2
Jing Chen 2
S. P. Glasby 2
Songtao Guo 2
Yutong Yu 2
Jie Wang 2
Zhengyu Gu 2
Mikko Korhonen 2
Ben Gong Lou 2
Dianjun Wang 2
Shaofei Du 2
Robert Jajcay 2
Qihang Liu 2
Anne Thomas 2

An O'Nan-Scott Theorem for Finite Quasiprimitive Permutation Groups and an Application to 2-Arc Transitive Graphs

1993-04-01
Cheryl E. Praeger
A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines … A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'Nan-Scott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The structure theorem for quasiprimitive groups is used to investigate the structure of quasiprimitive 2-arc transitive graphs, and a new construction is given for a family of such graphs.

The Magma Algebra System I: The User Language

1997-09-01
Wieb Bosma , John Cannon , Catherine Playoust
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On the full automorphism group of a graph

1981-09-01
Chris Godsil

A family of cubical graphs

1947-10-01
W. T. Tutte
We begin with some definitions. A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes. We begin with some definitions. A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.

The nonexistence of 8-transitive graphs

1981-09-01
Richard M. Weiss

The maximal factorizations of the finite simple groups and their automorphism groups

1990-01-01
Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl

Analysing finite locally 𝑠-arc transitive graphs

2003-08-25
Michael Giudici , Cai Heng Li , Cheryl E. Praeger
We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and are either locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G,s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–arc transitive for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given a normal subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> preserves both local primitivity and local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–arc transitivity and leads us to study graphs where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.

On Finite Affine 2-Arc Transitive Graphs

1993-09-01
А. А. Иванов , Cheryl E. Praeger

Finite Permutation Groups.

1966-08-01
Franklin Haimo , Helmut Wielandt

Permutation Groups

1996-01-01
John D. Dixon , Brian Mortimer

Automorphism groups and isomorphisms of Cayley digraphs

1998-03-01
Mingyao Xu

Subgroups of prime power index in a simple group

1983-04-01
Robert M. Guralnick

On the Symmetry of Cubic Graphs

1959-01-01
W. T. Tutte
Let G be a connected finite graph in which each edge has two distinct ends and no two distinct edges have the same pair of ends. We suppose further that … Let G be a connected finite graph in which each edge has two distinct ends and no two distinct edges have the same pair of ends. We suppose further that G is cubic , that is, each vertex is incident with just three edges. An s-path in G, where s is any positive integer, is a sequence S = (v 0 , v 1 … , v s ) of s + 1 vertices of G, not necessarily all distinct, which satisfies the following two conditions: (i) Any three consecutive terms of S are distinct. (ii) Any two consecutive terms of S are the two ends of some edge of G.
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The Finite Primitive Permutation Groups Containing an Abelian Regular Subgroup

2003-10-23
Cai Heng Li
A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside … A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside in 1900. 2000 Mathematics Subject Classification 20B15, 20B30

Finite transitive permutation groups and finite vertex-transitive graphs

1997-01-01
Cheryl E. Praeger , Cai Heng Li , Alice C. Niemeyer

The Affine Permutation Groups of Rank Three

1987-05-01
Martin W. Liebeck

Finite Quasiprimitive Graphs

1997-07-31
Cheryl E. Praeger
Summary A permutation group on a set Ω is said to be quasiprimitive on Ω if each of its nontrivial normal subgroups is transitive on Ω. For certain families of … Summary A permutation group on a set Ω is said to be quasiprimitive on Ω if each of its nontrivial normal subgroups is transitive on Ω. For certain families of finite arc-transitive graphs, those members possessing subgroups of automorphisms which are quasiprimitive on vertices play a key role. The manner in which the quasiprimitive examples arise, together with their structure, is described.

Finite Simple Groups

1982-01-01
Daniel Gorenstein

6-Transitive graphs

1980-04-01
Peter J‎. Cameron

The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$

2001-04-24
Cai Heng Li
A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 … A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text {M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text {PSp}(4,p)$ with $p$ prime and $p\equiv \pm 1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.
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A classification of the maximal subgroups of the finite alternating and symmetric groups

1987-12-01
Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl

On partitioning the orbitals of a transitive permutation group

2002-09-19
Cai Heng Li , Cheryl E. Praeger
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>be a permutation group on a set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"><mml:semantics><mml:mi mathvariant="normal">Ω</mml:mi><mml:annotation encoding="application/x-tex">\Omega</mml:annotation></mml:semantics></mml:math></inline-formula>with a transitive normal subgroup<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"><mml:semantics><mml:mi>M</mml:mi><mml:annotation encoding="application/x-tex">M</mml:annotation></mml:semantics></mml:math></inline-formula>. Then<inline-formula content-type="math/mathml"><mml:math … Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>be a permutation group on a set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"><mml:semantics><mml:mi mathvariant="normal">Ω</mml:mi><mml:annotation encoding="application/x-tex">\Omega</mml:annotation></mml:semantics></mml:math></inline-formula>with a transitive normal subgroup<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"><mml:semantics><mml:mi>M</mml:mi><mml:annotation encoding="application/x-tex">M</mml:annotation></mml:semantics></mml:math></inline-formula>. Then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>acts on the set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper O normal r normal b normal l left-parenthesis upper M comma normal upper Omega right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mi mathvariant="normal">b</mml:mi><mml:mi mathvariant="normal">l</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Orbl}(M,\Omega )</mml:annotation></mml:semantics></mml:math></inline-formula>of nontrivial<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"><mml:semantics><mml:mi>M</mml:mi><mml:annotation encoding="application/x-tex">M</mml:annotation></mml:semantics></mml:math></inline-formula>-orbitals in the natural way, and here we are interested in the case where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper O normal r normal b normal l left-parenthesis upper M comma normal upper Omega right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mi mathvariant="normal">b</mml:mi><mml:mi mathvariant="normal">l</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Orbl}(M,\Omega )</mml:annotation></mml:semantics></mml:math></inline-formula>has a partition<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation></mml:semantics></mml:math></inline-formula>such that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>acts transitively on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation></mml:semantics></mml:math></inline-formula>. The problem of characterising such tuples<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma upper G comma normal upper Omega comma script upper P right-parenthesis"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">(M,G,\Omega ,\mathcal P)</mml:annotation></mml:semantics></mml:math></inline-formula>, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue script upper P EndAbsoluteValue"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">|\mathcal P|</mml:annotation></mml:semantics></mml:math></inline-formula>is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue script upper P EndAbsoluteValue equals 2"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">|\mathcal P|=2</mml:annotation></mml:semantics></mml:math></inline-formula>exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>-actions on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"><mml:semantics><mml:mi mathvariant="normal">Ω</mml:mi><mml:annotation encoding="application/x-tex">\Omega</mml:annotation></mml:semantics></mml:math></inline-formula>and on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal P</mml:annotation></mml:semantics></mml:math></inline-formula>, and gives some construction methods for TODs.

Distance-Regular Graphs

2011-11-30
Andries E. Brouwer , Willem H. Haemers

Cyclic regular subgroups of primitive permutation groups

2002-01-17
Gareth A. Jones

Finite edge-transitive Cayley graphs and rotary Cayley maps

2006-05-09
Cai Heng Li
This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley … This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

Fintte two-arc transitive graphs admitting a suzuki simple group

1999-01-01
Xin Fang , Cheryl E. Preager
(1999). Fintte two-arc transitive graphs admitting a suzuki simple group. Communications in Algebra: Vol. 27, No. 8, pp. 3727-3754. (1999). Fintte two-arc transitive graphs admitting a suzuki simple group. Communications in Algebra: Vol. 27, No. 8, pp. 3727-3754.

Finite normal edge-transitive Cayley graphs

1999-10-01
Cheryl E. Praeger
An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are … An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G , each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G . Moreover, given a normal edge-transitive Cayley graph Γ H for a quotient group G / H , necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has Γ H as a homomorphic image, and a method for obtaining all such graphs Γ is given.
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Transitive Subgroups of Primitive Permutation Groups

2000-12-01
Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl

On Factorizations of Almost Simple Groups

1996-10-01
Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl

Finite Permutation Groups

1964-01-01
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Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

2004-07-26
Cai Heng Li
In this paper, a characterisation is given of finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s … In this paper, a characterisation is given of finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s\ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, it is shown that, for any given integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k not-equals 7"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\not =7</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a finite set (maybe empty) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s element-of StartSet 3 comma 4 comma 5 comma 7 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">s\in \{3,4,5,7\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transitive Cayley graphs of valency <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are their normal covers. This indicates that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc transitive Cayley graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s\ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

Regular subgroups of primitive permutation groups

2009-11-06
Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl
The authors address the classical problem of determining finite primitive permutation groups G with a regular subgroup B. The main theorem solves the problem completely under the assumption that G … The authors address the classical problem of determining finite primitive permutation groups G with a regular subgroup B. The main theorem solves the problem completely under the assumption that G is almost simple. While there are many examples of regular subgroups of small degrees, the list is rather short (just four infinite families) if the degree is assumed to be large enough, for example at least 30!. Another result determines all primitive groups having a regular subgroup which is almost simple. This has an application to the theory of Cayley graphs of simple groups.

On a reduction theorem for finite, bipartite, 2-arc transitive graphs

1993-01-01
Cheryl E. Praeger

Distance-Regular Graphs

1989-01-01
Andries E. Brouwer , Arjeh M. Cohen , Arnold Neumaier

Characterizing finite locally s-arc transitive graphs with a star normal quotient

2006-01-01
Michael Giudici , Cai Heng Li , Cheryl E. Praeger
Let Γ be a finite locally (G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of … Let Γ be a finite locally (G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G-orbits. In previous work the authors showed that if G has a non-trivial normal subgroup intransitive on both of the vertex orbits of G, then Γ is a cover of a smaller locally s-arc transitive graph. Thus the ‘basic’ graphs to study are those for which G acts quasiprimitively on at least one of the two orbits. In this paper we investigate the case where G is quasiprimitive on only one of the two G-orbits. Such graphs have a normal quotient which is a star. We construct several infinite families of locally 3-arc transitive graphs and prove characterization results for several of the possible quasiprimitive types for G.

Atlas of Finite Groups--Maximal Subgroups and Ordinary Characters for Simple Groups.

1987-01-01
R. Steinberg , J. H. Conway , Robert T. Curtis +3 more
This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections

Isomorphism Problem for Relational Structures with a Cyclic Automorphism

1987-01-01
Péter P. Pálfy

Factorisations of sporadic simple groups

2006-05-22
Michael Giudici

On the maximal subgroups of the finite classical groups

1984-10-01
Michael Aschbacher

The factorizations of the finite exceptional groups of lie type

1987-04-01
Christoph Hering , Martin W. Liebeck , Jan Saxl

Vertex-transitive graphs: Symmetric graphs of prime valency

1984-03-01
Peter Lorimer
Abstract Let G be a group acting symmetrically on a graph Σ, let G 1 be a subgroup of G minimal among those that act symmetrically on Σ, and let … Abstract Let G be a group acting symmetrically on a graph Σ, let G 1 be a subgroup of G minimal among those that act symmetrically on Σ, and let G 2 be a subgroup of G 1 maximal among those normal subgroups of G 1 which contain no member except 1 which fixes a vertex of Σ. The most precise result of this paper is that if Σ has prime valency p , then either Σ is a bipartite graph or G 2 acts regularly on Σ or G 1 | G 2 is a simple group which acts symmetrically on a graph of valency p which can be constructed from Σ and does not have more vertices than Σ. The results on vertex‐transitive groups necessary to establish results like this are also included.

On a Class of Doubly Transitive Groups: II

1964-05-01
Michio Suzuki
Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of … Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of the investigation is to classify the groups which do not contain a regular normal subgroup of order 1 + N in case N is even. (Author)

Finite Groups III

1982-01-01
Bertram Huppert , Norman Blackburn

On isomorphisms of connected Cayley graphs

1998-01-01
Li Cai Heng

The finite simple groups with at most two fusion classes of every order

1996-01-01
Cai Heng Li , Cheryl E. Praeger
Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class … Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class of a ϵ G is the set {aσ | σ ϵ Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where: (i) G has at most two fusion classes of order i for every i (23 examples); and (ii) any two elements of G of the same order are fused or inversenfused. The examples in case (ii) are: A5, A6,L2(7),L2(8), L3(4), Sz(8), M11 and M23An application is given concerning isomorphisms of Cay ley graphs.

Ádám's conjecture is true in the square-free case

1995-10-01
Mikhail Muzychuk

Vertex-Primitive Graphs of Order a Product of Two Distinct Primes

1993-11-01
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Recent developments in half-transitive graphs

1998-03-01
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Classifying Arc-Transitive Circulants

2004-11-01
István Kovács
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On finite groups all of whose elements of the same order are conjugate in their automorphism groups

1992-12-01
Zhang Ji-ping