We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in …
We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.
We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in …
We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.
We show that, in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits, and extensions is also singly-generated under isomorphisms and direct limits, and in …
We show that, in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits, and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective.In this way, we extend to the group-theoretic framework the topological analogue proved by Chachólski, Parent, and Stanley in 2004.We also establish several new relations between singly-generated closed classes.
A problem that arises in every group is the determination of the structure of all subgroups contained in it. Early investigations of the special case of free groups culminated in …
A problem that arises in every group is the determination of the structure of all subgroups contained in it. Early investigations of the special case of free groups culminated in the Reidemeister-Schreier theorem [1]2 which gives generators and relations for any subgroup of a group defined by generators and relations in terms of certain knowledge of its left cosets. In ?1, which collects together elementary definitions and results for use in the remainder of the paper, we show that the problem of Reidemeister and Schreier is exactly equivalent to that of giving free generators for a free group. We give a new solution to this problem (cf. Theorems 2.11, 2.25, and 2.27 of ?2) which permits application to questions of the structure of subgroups of groups which are free products or free products with one identified subgroup (?3, 4, 6). In particular, it provides an immediate proof of the Kurosh subgroup theorem [2] which states that every subgroup of a free product G is itself a free product, whose are a free group and subgroups of conjugates of the of G. The Kurosh proof, which is combinatorial in nature and utilizes a complicated double transfinite induction, was replaced by Baer and F. Levi with a topological proof [3] which represents a generalization of the second Johannson proof [4] of the ReidemeisterSchreier theorem. Their methods enable them to sharpen the theorem in an essential manner; namely, they provide us with a factorization with the largest possible factors which are subgroups of conjugates of of G. A further improvement of Kurosh's theorem was made by Takahasi [5] who established a relation between the number of of G and the number of of the subgroup. As proved in ?3, our statement of the Kurosh subgroup theorem includes all of these results, admitting in addition an explicit construction of generators for the subgroup. Contrary to all previous algebraic proofs, which have utilized the comparison of the length of a product of group elements with those of the in an essential manner, no cancellation arguments are made. Furthermore, although the statements are close to those of Takahasi, the removal of minimal conditions allows considerable freedom in the construction of generators. Recently, H. Neumann [6] has investigated theorems analogous to the Kurosh subgroup theorem for generalized free products with identified subgroups. By a simple improvement in our choice of generators, we obtain Theorem 4.08
This chapter is devoted to harmonic analysis on some compact abelian groups other than the circle group. We shall construct Haar measure for compact abelian groups, then prove the Pontryagin …
This chapter is devoted to harmonic analysis on some compact abelian groups other than the circle group. We shall construct Haar measure for compact abelian groups, then prove the Pontryagin duality theorem for compact and discrete abelian groups, a theorem of Minkowski, Kolmogorov’s extension theorem, and finally the Banach-Steinhaus theorem as a consequence of a theorem of Steinhaus about the set of distances between points in a set of positive measure.
We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.
We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.
This book constitutes the proceedings of a conference held at the University of Birmingham to mark the retirement of Professor A. M. Macbeath. The papers represent up-to-date work on a …
This book constitutes the proceedings of a conference held at the University of Birmingham to mark the retirement of Professor A. M. Macbeath. The papers represent up-to-date work on a broad spectrum of topics in the theory of discrete group actions, ranging from presentations of finite groups through the detailed study of Fuchsian and crystallographic groups, to applications of group actions in low dimensional topology, complex analysis, algebraic geometry and number theory. For those wishing to pursue research in these areas, this volume offers a valuable summary of contemporary thought and a source of fresh geometric insights.
A map $\mathcal{K}$ on a surface is called vertex-transitive if the automorphism group of $\mathcal{K}$ acts transitively on the set of vertices of $\mathcal{K}$. If the face\mbox{-}cycles at all the …
A map $\mathcal{K}$ on a surface is called vertex-transitive if the
automorphism group of $\mathcal{K}$ acts transitively on the set of vertices of
$\mathcal{K}$. If the face\mbox{-}cycles at all the vertices in a map are of
same type then the map is called semi\mbox{-}equivelar. In general,
semi\mbox{-}equivelar maps on a surface form a bigger class than
vertex-transitive maps. There are semi\mbox{-}equivelar toroidal maps which are
not vertex\mbox{-}transitive. A map is called minimal if the number of vertices
is minimal. A map $\mathcal{M} \to \mathcal{K}$ is a covering map if there is a
covering map from the vertex set of $\mathcal{M}$ to the vertex set of
$\mathcal{K}$. A covering map $f$ is a surjection and a local isomorphism
$\colon$ the neighbourhood of a vertex $v$ in $\mathcal{M}$ is mapped
bijectively onto the neighbourhood of $f(v)$ in $\mathcal{K}$. We know the bounds of number of vertex orbits of semi-equivelar toroidal
maps. These bounds are sharp. Datta \cite{BD2020} has proved that every
semi-equivelar toroidal map has a vertex-transitive cover, i.e., every
semi-equivelar toroidal map has a $1$-orbital semi-equivelar covering map. In
this article, we prove that if a semi-equivelar map is $k$ orbital then it has
a finite index $m$-orbital minimal covering map for $m \le k$. We also show the
existence and classification of $n$ sheeted covering maps of semi-equivelar
toroidal maps for each $n \in \mathbb{N}$.
If the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. In particular, it is called equivelar if the face-cycles contain …
If the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. In particular, it is called equivelar if the face-cycles contain same type of faces. A map is semiregular (or almost regular) if it has as few flag orbits as possible for its type. A map is $k$-regular if it is equivelar and the number of flag orbits of the map $k$ under the automorphism group. In particular, if $k =1$, its called regular. A map is $k$-semiregular if it contains more number of flags as compared to its type with the number of flags orbits $k$. Drach et al. \cite{drach:2019} have proved that every semi-equivelar toroidal map has a finite unique minimal semiregular cover. In this article, we show the bounds of flag orbits of semi-equivelar toroidal maps, i.e., there exists $k$ for each type such that every semi-equivelar map is $\ell$-uniform for some $\ell \le k$. We show that none of the Archimedean types on the torus is semiregular, i.e., for each type, there exists a map whose number of flag orbits is more than its type. We also prove that if a semi-equivelar map is $m$-semiregular then it has a finite index $t$-semiregular minimal cover for $t \le m$. We also show the existence and classification of $n$ sheeted $k$-semiregular maps for some $k$ of semi-equivelar toroidal maps for each $n \in \mathbb{N}$.
A map $K$ on a surface is called edge-transitive if the automorphism group of $K$ acts transitively on the set of edges of $K$. A tiling is edge-homogeneous if any …
A map $K$ on a surface is called edge-transitive if the automorphism group of $K$ acts transitively on the set of edges of $K$. A tiling is edge-homogeneous if any two edges with vertices of congruent face-cycles. In general, edge-homogeneous maps on a surface form a bigger class than edge-transitive maps. There are edge-homogeneous toroidal maps which are not edge\mbox{-}transitive. A map is called minimal if the number of edges is minimal. A map $f \colon M \to K$ is a covering map if $f$ is a covering map from the vertex set of $M$ to the vertex set of $K$ such that $f$ is a surjection and a local isomorphism $\colon$ the neighbourhood of a vertex $v$ in $M$ is mapped bijectively onto the neighbourhood of $f(v)$ in $K$. Orbani{\' c} et al. and {\v S}ir{\'a}{\v n} et al. have shown that every edge-homogeneous toroidal map has edge-transitive cover. In this article, we show the bounds of edge orbits of edge-homogeneous toroidal maps. We also prove that if a edge-homogeneous map is $k$ edge orbital (or $k$ orbital) then it has a finite index $m$-edge orbital minimal cover for $m \le k$. We also show the existence and classification of $n$ sheeted covers of edge-homogeneous toroidal maps for each $n \in \mathbb{N}$.
A map $\mathcal{K}$ on a surface is called vertex-transitive if the automorphism group of $\mathcal{K}$ acts transitively on the set of vertices of $\mathcal{K}$. If the face\mbox{-}cycles at all the …
A map $\mathcal{K}$ on a surface is called vertex-transitive if the automorphism group of $\mathcal{K}$ acts transitively on the set of vertices of $\mathcal{K}$. If the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. In general, semi\mbox{-}equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi\mbox{-}equivelar toroidal maps which are not vertex\mbox{-}transitive. A map $\mathcal{M} \to \mathcal{K}$ is a covering map if there is a covering map from the vertex set of $\mathcal{M}$ to the vertex set of $\mathcal{K}$. A covering map $f$ is a surjection and a local isomorphism $\colon$ the neighbourhood of a vertex $v$ in $\mathcal{M}$ is mapped bijectively onto the neighbourhood of $f(v)$ in $\mathcal{K}$.
We know the bounds of number of vertex orbits of semi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} has proved that every semi-equivelar toroidal map has vertex-transitive cover, i.e., every semi-equivelar toroidal map has a $1$-orbital semi-equivelar covering map. In this article, we prove that if a semi-equivelar map is $k$ orbital then it has a finite index $m$-orbital covering map for $m \le k$.
The first portrait of a group was given in Burnside [1]. Burnside constructed a portrait of the cyclic group with n elements and the free group, Fn, on n generators …
The first portrait of a group was given in Burnside [1]. Burnside constructed a portrait of the cyclic group with n elements and the free group, Fn, on n generators in the Euclidean plane. More precisely, since the one to one plane transformations that Burnside used were inversions in a circle, they should be thought of as transformations of the Riemann sphere. The relationship between the circles used determines the resulting group. The construction of the free group, F2, can be most easily drawn in the hyperbolic plane (see Burnside [1], page 379). Burnside began with a single region associated with the identity transformation, E. Since inversion in a circle reverses the orientation of the plane, Burnside used a composition of two inversions for each element of the free group. The region E is colored white and the region obtained by a single inversion is colored black. Therefore, a fundamental region for the group of transformations is the union of one white and one black region. Each white region (and its associated black region) is labeled with the element of the free group that transforms the fundamental region into that region. Figure 1 is a portrait of this two generator free group constructed by Geometer's SketchPad. Each 'triangle' is bounded by arcs colored red, blue or black in our sketch. Inversion takes a shaded region into a non-shaded region and vice versa. Therefore, each group action is represented by the composite of two such inversions. We interpret this picture as a portrait of a group with presentation 2 1 | , , F uvw w v u ≅ = .
A pseudofree group action on a space X is one whose set of singular orbits forms a discrete subset of its orbit space.Equivalently -when G is finite and X is …
A pseudofree group action on a space X is one whose set of singular orbits forms a discrete subset of its orbit space.Equivalently -when G is finite and X is compact -the set of singular points in X is finite.In this paper, we classify all of the finite groups which admit pseudofree actions on S 2 × S 2 .The groups are exactly those that admit orthogonal pseudofree actions on S 2 × S 2 ⊂ ޒ 3 × ޒ 3 , and they are explicitly listed.This paper can be viewed as a companion to a preprint of Edmonds, which uniformly treats the case in which the second Betti number of a fourmanifold M is at least three.
(1981). An Elementary Course in Mathematical Symmetry. The American Mathematical Monthly: Vol. 88, No. 1, pp. 59-64.
(1981). An Elementary Course in Mathematical Symmetry. The American Mathematical Monthly: Vol. 88, No. 1, pp. 59-64.
There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of …
There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A commutative set of differential-difference operators, each homogeneous of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.
Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We …
Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid ; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K.
This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the $1+1+16=18$ families of finite simple …
This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the $1+1+16=18$ families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated "pariah" groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group $\mathbb M$, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the $5+7+8+6=26$ sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.
Abstract We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the …
Abstract We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the growth rates of planar hyperbolic Coxeter groups are spectral radii of Coxeter transformations, and show that this need not be the case for growth rates of hyperbolic tetrahedral Coxeter groups.
Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> G_{0} be the stabilizer …
Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> G_{0} in Ω, including the trivial orbit <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{\alpha\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.