In this paper, we classify finite permutation groups with a transitive abelian subgroup that are almost simple, quasiprimitive and innately transitive, which extend the results of Li and Praeger that …
In this paper, we classify finite permutation groups with a transitive abelian subgroup that are almost simple, quasiprimitive and innately transitive, which extend the results of Li and Praeger that is on finite permutation groups with a transitive cyclic subgroup.
Two problems in the theory of finite permutation groups are considered in this thesis: A. transitive groups of degree p, where p = 4q+1 and p,q are prime, B. …
Two problems in the theory of finite permutation groups are considered in this thesis: A. transitive groups of degree p, where p = 4q+1 and p,q are prime, B. automorphism groups of 2-graphs and some related algebras. Problem A should be seen in the following context: in 1963. N.Ito began a study of insoluble, transitive groups G of degree p on a set Ω, where p = 2q+1 and p,q are prime, showing among other things, that such a group G is 3-transitive. His methods involve the modular character theory of G for both the primes p and q (developed by R.Brauer). He uses this theory to prove facts about the permutation characters of G associated with Ω (2) and Ω {2} , the sets of ordered and unordered pairs (respectively) of distinct elements of Ω. The first part of this thesis represents an attempt to extend these methods to the case p = 4q+1. The main result obtained is Theorem. Let G be an insoluble, transitive permutation group of degree p, where p = 4q+1 and p.q are prime with p>13. Then G is 3-transitive. Also some progress is made towards a proof that the groups in Problem A are 4-transitive. In the second part of this thesis (Problem B) certain algebras are defined from 2-graphs as follows: let (Ω,Δ) be a 2-graph, that is, Δ is a set of 3-subsets of a finite set Ω such that every 4-subset of Ω contains an even number of elements of Δ. Write Ω= {e 1 ....,e n }. Given any field F of characteristic 2, make FΩ into an algebra by defining [see text for continuation of abstract].
Let G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We …
Let G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.
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
A finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. In this paper, we give a characterisation and structure theorem for the …
A finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. In this paper, we give a characterisation and structure theorem for the finite innately transitive groups, as well as describing those innately transitive groups which preserve a product decomposition. The class of innately transitive groups contains all primitive and quasiprimitive groups. 2000 Mathematics Subject Classification 20B05, 20B15.
We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.
We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.
We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.
We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.
This dissertation focuses on investigating Schurian association schemes, which are those that are induced by a group acting transitively on a set. Specifically, we focus on the n-dimensional unitary group …
This dissertation focuses on investigating Schurian association schemes, which are those that are induced by a group acting transitively on a set. Specifically, we focus on the n-dimensional unitary group over the finite field of order q^2, denoted GU(n,q), acting on geometric sets coming from unitary space, all of an isotropic flavor. In Chapter 1, we provide the necessary background, not only pertaining to association schemes themselves, but also in regard to the needed field theory, results about unitary geometry, and connections to graph theory. Chapter 2 provides a thorough investigation of the family of Schurian association schemes resulting from the group action of GU(n,q) acting on the set of isotropic vectors, Omega(n,q), that is, those nonzero vectors whose Hermitian inner product is 0. We begin by looking at the group action itself, then continue by computing the intersection matrices for all values of n and q, and finish by giving the character table for the only commutative case, namely when q=2. The remainder of the thesis concentrates around totally isotropic subspaces of unitary space, starting with the 1-dimensional subspaces in Chapter 3, whose character tables are completely determined, providing a connection to some other results that have already been found. We again produce the intersection matrices and character tables for all values of n and q. Chapter 2 proceeds to the 2-dimensional case, in which partial results can be found for the intersection matrices and character tables for n=4 and n=5, but the relations are found for general n and q still. Finally, a generalization to the d-dimensional case is discussed in Chapter 5, describing the relations of this family of association schemes and the rank, as well as a method to find intersection matrices and character tables for restricted values of n.
In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true …
In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true for all sequences of trees that are 'almost stars'; and we prove that the Erdos-Sos conjecture is true for all graphs G with girth at least 5. We also conjecture that every graph G with minimal degree k and girth at least $2t+1$ contains every tree T of order $kt+1$ such that $\Delta(T)\leq k.$ This conjecture is trivially true for t = 1. We Prove the conjecture is true for t = 2 and that, for this value of t, the conjecture is best possible. We also provide supporting evidence for the conjecture for all other values of t. In algebraic graph theory, we are primarily concerned with isomorphism problems for vertex-transitive graphs, and with calculating automorphism groups of vertex-transitive graphs. We extend Babai's characterization of the Cayley Isomorphism property for Cayley hypergraphs to non-Cayley hypergraphs, and then use this characterization to solve the isomorphism problem for every vertex-transitive graph of order pq, where p and q distinct primes. We also determine the automorphism groups of metacirculant graphs of order pq that are not circulant, allowing us to determine the nonabelian groups of order pq that are Burnside groups. Additionally, we generalize a classical result of Burnside stating that every transitive group G of prime degree p, is doubly transitive or contains a normal Sylow p-subgroup to all $p\sp k,$ provided that the Sylow p-subgroup of G is one of a specified family. We believe that this result is the most significant contained in this dissertation. As a corollary of this result, one easily gives a new proof of Klin and Poschel's result characterizing the automorphism groups of circulant graphs of order $p\sp k,$ where p is an odd prime.
In this survey paper I will talk about the classification of the maximal subgroups of the symmetric group of degree n, which can be divided into three main categories: intransitive, …
In this survey paper I will talk about the classification of the maximal subgroups of the symmetric group of degree n, which can be divided into three main categories: intransitive, imprimitive and primitive maximal subgroups.The O'Nan-Scott theorem and its proof will be discussed.The content of this paper was inspired by two courses I gave, one was a minicourse given in December 2018 in the University of Campinas (state of S.Paulo, Brazil) during the brazilian Escola de Álgebra 2018, the other was an online PhD course given in November 2020 for the University of Padova (Italy).This paper is structured as lecture notes, in particular it contains exercises for the reader.
The purpose of this note is to give a new proof of a theorem of L. Carlitz [ 2 ] and R. McConnel [ 5 ]. The theorem is as …
The purpose of this note is to give a new proof of a theorem of L. Carlitz [ 2 ] and R. McConnel [ 5 ]. The theorem is as follows: THEOREM 1. Let F = GF(p n ) be the finite field of order q = p n and let K — {x ∈ F|x d = 1} for some proper divisor d of q — 1 .
Any Schur ring is uniquely determined by a partition of the elements of the group. In this paper we present a general technique for enumerating Schur rings over cyclic groups …
Any Schur ring is uniquely determined by a partition of the elements of the group. In this paper we present a general technique for enumerating Schur rings over cyclic groups using traditional Schur rings. We also survey recent efforts to enumerate Schur rings over cyclic groups of specific orders.
A Cayley (di)graph Cay(G,S) of a group G is called normal if the right regular representation of G is normal in the full automorphism group of Cay(G,S), and a CI-(di)graph …
A Cayley (di)graph Cay(G,S) of a group G is called normal if the right regular representation of G is normal in the full automorphism group of Cay(G,S), and a CI-(di)graph if for every Cayley (di)graph Cay(G,T),Cay(G,S)≅Cay(G,T) implies that there is σ∈Aut(G) such that Sσ=T. We call a group G an NDCI-group or NCI-group if all normal Cayley digraphs or graphs of G are CI-digraphs or CI-graphs, respectively. We prove that a cyclic group of order n is an NDCI-group if and only if 8∤n, and an NCI-group if and only if either n = 8 or 8∤n.