CHAPTER IV - Applications of the Theory of Characters

An infinite group G is called just infinite if all non-trivial normal subgroups have finite index; if G is profinite it is merely required that all non-trivial closed normal subgroups have finite index. Just infinite groups have arisen in a variety of contexts. The abstract just infinite groups having non-trivial abelian normal subgroups are precisely the space groups whose point groups act rationally irreducibly on the abelian normal subgroups (see McCarthy [7]). Many arithmetic groups are known to be just infinite modulo their centres; examples are SLn (R) for n ≥ 3 and Sp2n (R) for n ≥ 2, where R is the ring of integers of an algebraic number field (see [1]). The Nottingham group over \(\mathop \mathbb{F}\nolimits_p\) (described in Chapter 6) is a just infinite pro-p Groups. Grigorchuk [3], [4] and Gupta and Sidki [5] have introduced and studied some infinite finitely generated p-groups which act on trees, and many, together with their pro-p completions, are just infinite. Using Zorn’s Lemma it is easy to see that if S is either an infinite finitely generated abstract group or an infinite finitely generated pro-p group, then S has a just infinite quotient group. Therefore to decide whether a group-theoretic property implies finiteness, it is sometimes sufficient to consider just infinite groups.KeywordsNormal SubgroupWreath ProductFinite IndexMaximal ConditionBoolean LatticeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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. Introduction 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. This is an essay about families of finite arc-transitive graphs which have group-theoretic defining properties. By a quasiprimitive graph in such a family we shall mean a graph which admits a subgroup of automorphisms which not only is quasiprimitive on vertices, but also has the defining property of the family. For example, in the family of all arc-transitive graphs, a quasiprimitive graph is one with a subgroup of automorphisms which is both quasiprimitive on vertices and transitive on arcs. (An arc of a graph Γ is an ordered pair of adjacent vertices.) First we shall describe an approach to studying several families of finite arctransitive graphs whereby quasiprimitive graphs arise naturally. The concept of quasiprimitivity is a weaker notion than that of primitivity for permutation groups, and we shall see that finite quasiprimitive permutation groups may be described in a manner analogous to the description of finite primitive permutation groups provided by the famous theorem of M. E. O'Nan and L. L. Scott [17, 30]. There are several distinct types of finite quasiprimitive permutation groups, and several corresponding distinct types of finite quasiprimitive graphs.

<!-- *** Custom HTML *** --> This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups. Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen–Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers. Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space $C(G,\mathbb C)$ of all complex-valued functions on a finite group $G$. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside's Theorem as an application of this theory—that any finite group of order $p^aq^b$ with $p$ and $q$ prime and with $a+b&gt;1$ has a nontrivial normal subgroup. Section 6 introduces cohomology of groups in connection with group extensions. If $N$ is to be a normal subgroup of $G$ and $Q$ is to be isomorphic to $G/N$, the first question is to parametrize the possibilities for $G$ up to isomorphism. A second question is to parametrize the possibilities for $G$ if $G$ is to be a semidirect product of $N$ and $Q$.

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.

Finite locally-quasiprimitive graphs

Quotients and inclusions of finite quasiprimitive permutation groups

Let $H$ be a subgroup of a group $G$. A normal subgroup $N_H$ of $H$ is said to be inheritably normal if there is a normal subgroup $N_G$ of $G$ such that $N_H=N_G\cap H$. It is proved in the paper that a subgroup $N_{G_i}$ of a factor $G_i$ of the $n$-periodic product $\prod_{i\in I}^nG_i$ with nontrivial factors $G_i$ is an inheritably normal subgroup if and only if $N_{G_i}$ contains the subgroup $G_i^n$. It is also proved that for odd $n\ge 665$ every nontrivial normal subgroup in a given $n$-periodic product $G=\prod_{i\in I}^nG_i$ contains the subgroup $G^n$. It follows that almost all $n$-periodic products $G=G_1\overset{n}{\ast}G_2$ are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.

Groups without Faithful Transitive Permutation Representations of Small Degree

A permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

Quasiprimitivity: structure and combinatorial applications

On primitive overgroups of quasiprimitive permutation groups

Arc-transitive digraphs with quasiprimitive local actions

Symmetry adapted functions belonging to the crystallographic groups