On the universality of systems of words in permutation groups

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.

Until 1980, there was no such subject as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I shall concentrate on a few topics, following the pattern of my conference lectures: the random graph (a case study); homogeneous relational structures (a powerful construction technique for interesting permutation groups); oligomorphic permutation groups (where the relations with other areas such as logic and combinatorics are clearest, and where a number of interesting enumerative questions arise); and the Urysohn space (another case study). I have preceded this with a short section introducing the language of permutation group theory, and I conclude with briefer accounts of a couple of topics that didn't make the cut for the lectures (maximal subgroups of the symmetric group, and Jordan groups). I have highlighted a few specific open problems in the text. It will be clear that there are many wide areas needing investigation! I have also included some additional references not referred to in the text. Notation and terminology This section contains a few standard definitions concerning permutation groups. I write permutations on the right: that is, if g is a permutation of a set Ω, then the image of α under g is written α g .

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.

A subgroup H is called c -normal in a group G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G , where H G ≕ Core( H ) is the maximal normal subgroup of G which is contained in H . We obtain the c -normal subgroups in symmetric and dihedral groups. Also we find the number of c -normal subgroups of order 2 in symmetric groups. We conclude by giving a program in GAP for finding c -normal subgroups. AMS Classification : 20D25. Keywords : c -normal, symmetric, dihedral. Introduction The relationship between the properties of maximal subgroups of a finite group G and the structure of G has been studied extensively. The normality of subgroups in a finite group plays an important role in the study of finite groups. It is well known that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G . In Wang introduced the concept of c -normality of a finite group. He used the c -normality of a maximal subgroup to give some conditions for the solvability and supersolvability of a finite group. For example, he showed that G is solvable if and only if M is c -normal in G for every maximal subgroup M of G . In this paper, we obtain the c -normal subgroups in symmetric and dihedral groups, and also we find the number of c -normal subgroups of order 2 in symmetric groups.

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

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-

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.

CHAPTER IV - Applications of the Theory of Characters

In 1999 a number of eminent mathematicians were invited to Bielefeld to present lectures at a conference on topological, combinatorial and arithmetic aspects of (infinite) groups. The present volume consists of survey and research articles invited from participants in this conference. Topics covered include topological finiteness properties of groups, Kac-Moody groups, the theory of Euler characteristics, the connection between groups, formal languages and automata, the Magnus-Nielsen method for one-relator groups, atomic and just infinite groups, topology in permutation groups, probabilistic group theory, the theory of subgroup growth, hyperbolic lattices in dimension three, generalised triangle groups and reduction theory. All contributions are written in a relaxed and attractive style, accessible not only to specialists, but also to good graduate and post-graduate students, who will find inspiration for a number of basic research projects at various levels of technical difficulty.

A group $$A$$ A acting faithfully on a set $$X$$ X is $$2$$ 2 -distinguishable if there is a $$2$$ 2 -coloring of $$X$$ X that is not preserved by any nonidentity element of $$A$$ A , equivalently, if there is a proper subset of $$X$$ X with trivial setwise stabilizer. The motion of an element $$a \in A$$ a ? A is the number of points of $$X$$ X that are moved by $$a$$ a , and the motion of the group $$A$$ A is the minimal motion of its nonidentity elements. When $$A$$ A is finite, the Motion Lemma says that if the motion of $$A$$ A is large enough (specifically at least $$2\log _2 |A|$$ 2 log 2 | A | ), then the action is $$2$$ 2 -distinguishable. For many situations where $$X$$ X has a combinatorial or algebraic structure, the Motion Lemma implies that the action of $$\mathrm{Aut }(X)$$ Aut ( X ) on $$X$$ X is 2-distinguishable in all but finitely many instances. We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee $$2$$ 2 -distinguishability. From this, we deduce a number of results, including the fact that every locally finite, connected graph whose automorphism group is countably infinite is $$2$$ 2 -distinguishable. One cannot extend the Motion Lemma to uncountable permutation groups, but nonetheless we prove that (under the permutation topology) every closed permutation group with infinite motion has a dense subgroup which is $$2$$ 2 -distinguishable. We conjecture an extension of the Motion Lemma which we expect holds for a restricted class of uncountable permutation groups, and we conclude with a list of open questions. The consequences of our results are drawn for orbit equivalence of infinite permutation groups.

Let G be a finite group with k conjugacy classes, and S(∞) be the infinite symmetric group, i.e., the group of finite permutations of {1, 2, 3, …}. Then the wreath product G ∞ = G ∼ S (∞) of G with S (∞) (called the big wreath product) can be defined. The group G ∞ is a generalization of the infinite symmetric group, and it is an example of a “big” group, in Vershik’s terminology. For such groups the two-sided regular representations are irreducible, the conventional scheme of harmonic analysis is not applicable, and the problem of harmonic analysis is a nontrivial problem with connections to different areas of mathematics and mathematical physics. Harmonic analysis on the infinite symmetric group was developed in the works by Kerov, Olshanski, and Vershik, and Borodin and Olshanski. The goal of this paper is to extend this theory to the case of G ∞. In particular, we construct an analogue $${\frak{S}}_{G}$$ S G of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on $${\frak{S}}_{G}$$ S G . Next, we introduce generalized regular representations $$\{T_{z_{1}},\cdots,z_{k} : z_{1}\in \mathbb{C},\cdots,z_{k}\in \mathbb{C}\}$$ { T z 1 , ⋯ , z k : z 1 ∈ C , ⋯ , z k ∈ C } of the big wreath product G ∞, which are analogues of the Kerov–Olshanski–Vershik generalized regular representations of the infinite symmetric group. We derive an explicit formula for the characters of $$T_{z_{1}},\cdots,z_{k}$$ T z 1 , ⋯ , z k . The spectral measures of these representations are characterized in different ways. In particular, these spectral measures are associated with point processes whose correlation functions are explicitly computed. Thus, in representation-theoretic terms, the paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of $$T_{z_{1}},\cdots,z_{k}$$ T z 1 , ⋯ , z k into irreducible components.

This is a survey of work by many people on various notions of largeness for subgroups of infinite symmetric groups. The primary concern is with maximal subgroups of infinite symmetric groups, of which several new examples are given. Subgroups of small index in symmetric groups (and in other permutation groups) are considered, as are questions about covering symmetric groups with families of subgroups. Examples are given of maximal subgroups of other large permutation groups, such as GL(κ, F) (where κ is an infinite cardinal) and Aut(Q,≤ ). The paper concludes with a discussion of other notions of largeness in symmetric groups, such as oligomorphic, Jordan, maximal closed, and various strong transitivity conditions on infinite subsets.

Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan–Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

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 \({\cal Z}\).