Andrews-Curtis groups

On graph-restrictive permutation groups

A transitive permutation group is called semiprimitive if each of its normal subgroups is either semiregular or transitive. The class of semiprimitive groups properly includes primitive groups, quasiprimitive groups and innately transitive groups. The latter three classes of rank 3 permutation groups have been classified, making significant progress towards solving the long-standing problem of classifying permutation groups of rank 3. In this paper, we complete the classification of finite semiprimitive groups of rank 3, building on the recent work of Huang, Li and Zhu. Examples include Schur coverings of certain almost simple 2-transitive groups and three exceptional small groups.

This chapter opens Part Two, devoted to the study of transformation theory, with some additional topics in group theory that arise in musical applications. Transformation groups on finite spaces may be regarded as permutation groups; permutation groups on pitch-class space include not only the groups of transpositions and inversions but also the multiplication group, the affine group, and the symmetric group. Another musical illustration of permutations involves the rearrangement of lines in invertible counterpoint. The structure of a finite group may be represented in the form of a group table or a Cayley diagram (a kind of graph). Other concepts discussed include homomorphisms and isomorphisms of groups, direct-product groups, normal subgroups, and quotient groups. Groups underlie many examples of symmetry in music, as formalized through the study of equivalence relations, orbits, and stabilizers.

CHAPTER IV - Applications of the Theory of Characters

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B u Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B u Aut(Ω) has \(2^{2^{\aleph_0}}\) normal subgroups, and there is a certain ideal \({\cal Z}\) of the lattice of subsets of Open image in new window (\(\mathbb{Z}\)), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to \({\cal Z}\).

A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970's. It is shown that a connected arc transitive circulant ? of order n is one of the following: a complete graph Kn, a lexicographic product $\Sigma [{\bar K}_b]$ , a deleted lexicographic product $\Sigma [{\bar K}_b] - b\Sigma$ , where ? is a smaller arc transitive circulant, or ? is a normal circulant, that is, Auta? has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.

In this paper, we classify 2-closed (in Wielandt's sense) permutation groups which contain a normal regular cyclic subgroup and prove that for each such group $G$, there exists a circulant $\Gamma$ such that $\mathrm{Aut} (\Gamma)=G$.

In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, we determine those groups which are equivalent in the sense that they may not be distinguished by the solvability of a system of finitely many equations in variables and parameters.

Introduction to group theory

A theory of semiprimitive groups

The process of imbedding a group in a larger group of some prescribed type has been one of the most useful tools in the investigation of properties of groups. The three principal types of representation of groups, each with its particular field of usefulness, are the following: 1. Permutation groups. 2. Monomial groups. 3. Linear or matrix representations of groups. These three types of representation correspond to an imbedding of the group in the following groups: 1. The symmetric group. 2. The complete monomial group. 3. The full linear group. The symmetric group and the full linear group have both been exhaustively investigated and many of their principal properties are known. A similar study does not seem to exist for the complete monomial group. Such a general theory seems particularly desirable in view of the numerous recent investigations on finite groups in which the monomial representations are used in one form or another to obtain deep-lying theorems on the properties of such groups. The present paper is an attempt to fill this lacuna. In this paper the monomial group or symmetry is taken in the most general sense(') where one considers all permutations of a certain finite number of variables, each variable being multiplied also by some element of a fixed arbitrary group H. In the first chapter the simplest properties such as transformation, normal form, centralizer, etc., are discussed. Some of the auxiliary theorems appear to have independent interest. One finds that the symmetry contains a normal subgroup, the basis group, consisting of all those elements which do not permute the variables. The symmetry splits over the basis group with a group isomorphic to the symmetric group as one representative group. A complete solution of the problem of finding all representative groups in this splitting of the symmetry is given. This result is of interest since it gives a general idea of the solution of the splitting problem in a fairly complicated case. In the second chapter all normal subgroups of the symmetry are deter-

Finite locally-quasiprimitive graphs

On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

A permutation group is cofinitary if any non-identity element fixes only finitely many points. This paper presents a survey of such groups. The paper has four parts. Sections 1–6 develop some basic theory, concerning groups with finite orbits, topology, maximality, and normal subgroups. Sections 7–12 give a variety of constructions, both direct and from geometry, combinatorial group theory, trees, and homogeneous relational structures. Sections 13–15 present some generalisations of sharply k-transitive groups, including an orbit-counting result with a character-theoretic flavour. The final section treats some miscellaneous topics. Several open problems are mentioned.

We investigate some situation in which automorphisms of a groupG are unique- ly determined by their restrictions to a proper subgroup H. Much of the paper is devoted to studying under which additional hypotheses this property forces G to be nilpotent if H is. As an application we prove that certain countably infinite locally nilpotent groups have uncountably many (outer) automorphisms. Homomorphisms of groups are defined by their restriction to any generating set of their domain. This property actually characterizes generating subsets of groups, for if H is a proper subgroup of the group G then there are two different homomorphisms from G to the same group K whose restrictions to H are the same: a simple construction due to Eilenberg and Moore is suggested as Exercise 3.35 in (10), p.54. The situation can be quite different if—rather than referring to all homomorphisms of domain a group G—we restrict attention to, say, endomorphisms of G only. For instance, if G is isomorphic to the rational group Q and g is any nontrivial element of G, then two endomorphisms of G coincide if they agree on g, in other words endomorphisms of G are uniquely determined by their restrictions to {g}. This suggests the following definition. Let G be a group and let be a set of endomorphisms of G. We say that a subset X of G is a -basis if and only if, for every �,� ∈ , it holds � = � if �|X = �|X, where �|X and �|X denote restrictions to X. We shall also use expressions like 'End- basis', 'Aut-basis' or 'Inn-basis of G' as synonym with EndG-, AutG-, or InnG-basis respectively. For instance, the above example can be rephrased by saying that in the rational group every one- element subset different from the identity subgroup is an End-basis. More generally, every maximal independent subset of a torsion-free abelian group A is an End-basis of A. The Aut-bases of a group G are just the bases of AutG viewed as a permutation group on G, whence the name. Indeed, it is clear that for any ≤ AutG a subset X of G is a -basis if and only if C (X) = 1. In particular, the Inn-bases of a group G are the subsets X of G such that CG(X) = Z(G). Other self-evident facts about -bases (for a set of endomorphisms of a group G) are that if X is a -basis then X1 is a 1-basis for any subset 1 of and any superset X1 of X contained in G. Also, X is a -basis if and only if h Xi is a -basis, so the property of being a -basis could be equivalently defined as an embedding property for subgroups. In this paper we shall assume this point of view and discuss some instances of the general problem of when a group G inherits group theoretical properties from a subgroup of G which is a -basis, for some specific ⊆ EndG. For example, it is immediate to observe that a group is abelian if it has an abelian subgroup which is an Inn-basis (Lemma 1.7). We will be mainly concerned with the case = AutG. A drastically restrictive result is that a direct power of every centreless group can be embedded as a normal subgroup which is an Aut-basis in a group with rather arbitrary structure (see Corollary 1.5). This is the reason why we turn our attention to group classes without nontrivial centreless groups, and mainly study nilpotent (sub)groups.