Ischia Group Theory 2008

The following sections are included:

• Preface

• Sponsors

• Conference Program

• Registered Participants

• Contents

In this survey we show the influence of the abnormal maximal subgroups of finite groups in their structure.

We present an introduction to the theory of Finite Gelfand Pairs and to their application to the study of the asymptotic behaviour of some Markov chains (the Bernoulli-Laplace diffusion model). We shall also present some new examples arising from Geometric Group Theory (self-similar groups, branch groups, the Basilica group, iterated monodromy groups) and the Theory of Association Schemes (generalized wreath products of permutation groups).

Imposing identical assumptions on characters and on conjugacy classes of finite groups, yields in many cases analogous results. Sometimes the conclusions of the analogs are (almost) identical as well. We call these, close analogs. We review (some old, some very elementary) examples of close analogs and describe a recent one which is an analog of a congruence of Blichfeldt involving the group order.

Let Ω be a set. Then an unordered relation on Ω is a family R of subsets of Ω. For such a family let be the group of all permutations g of Ω for which x^g belongs to R whenever x belongs to R. We are interested in permutation groups which can be represented in the shape for an unordered relation R on Ω.

In this paper we study infinite groups in which the polycyclicity is a transitive relation on the set of all 2-generator subgroups. We also present a short overview of known results about finite and infinite groups in which a given property is transitive on the set of all 2-generator subgroups.

This paper represents the content of a talk given by the first author at the Ischia Group Theory Conference, 2008.

Undirected profinite graphs are defined and, for any connected such graph, its fundamental group.

The purpose of this paper is investigating the structure of groups in which generating a cyclic subgroup is a transitive relation on the set of all non-identity elements. We will introduce this property from a condition on the diameter of the cyclic graph associated with a group G, that is the graph in which the set of vertices is the set of all pairs of non-identity elements of G and two distinct vertices x and y are joined if they generate a cyclic subgroup. We say that a group G is cyclic transitive if the condition that 〈x, y〉 and 〈y, z〉 are both cyclic implies that 〈x, z〉 is cyclic, where x, y, z are non-identity elements of G.

We study the influence of this condition of the structure of the group in the following cases: abelian, more generally nilpotent and finally supersolvable.

We consider here only finite p-groups. A p-group H is minimal nonabelian if and only if H = 〈a,b〉 is two-generated and |H′| = p, i.e., [a,b]^p = [a,b,a] = [a,b,b] = 1. In that case Φ(H) = Z(H) so that |H : Z(H)| = p².

It is well known that a nonabelian p-group is generated by its minimal nonabelian subgroups. We have proved a more precise result in Janko (⁴ Lemma 2.1).

Starting with an infinite cardinal κ₀, one may define a transfinite sequence {κ_ν} of cardinals with κ_ν+1 := 2^κ_ν and κ_λ := sup{κ_ν;ν < λ} for limit ordinals λ. Then, embedding the symmetric group S_ν+1 := Sym(κ_ν) into the symmetric group Sym (2^κ_ν+1) by the regular representation and defining S_λ to be the direct limit of the groups S_ν for ν < λ when λ is a limit ordinal, one obtains an ascending sequence {S_ν} of groups indexed by the class of ordinals. For a limit ordinal λ the group S_λ has many interesting properties. We show that it is a simple group, that isomorphic finitely generated subgroups are conjugate, that it has a remarkable factorisation, that it is universal, and that it is existentially closed.*

We discuss recent results in which a normal subgroup of finite index (or with finite rank of the quotient, or some other well-behaving parameter) satisfying an outer (multilinear) commutator law is transformed into a large characteristic subgroup satisfying the same law. Here "large" means that the index (or the corresponding parameter) of the resulting characteristic subgroup is finite and bounded in terms of the index (or the corresponding parameter) of the original normal subgroup. Similar results also hold for ideals in arbitrary algebras over a field and, moreover, for a wider class of algebraic systems defined in terms of multioperator groups. We also give examples of applications of these results in various situations involving almost soluble groups, in particular, in the study of groups with almost fixed-point-free automorphisms.

Let V be a vector space over the field F. If G is a subgroup of GL(V, F), then we define the central dimension of G (denoted by centdim_FG) as the F–dimension of the factor-space V/C_V(G). We say that G has finite central dimension if centdim _FG is finite and G has infinite central dimension otherwise. In this paper we study soluble linear groups such that the set of infinite central dimensional subgroups satisfies the weak minimal condition continuing the study started in [12,13,18].

We say a finite group G is a VZ-group if all its nonlinear irreducible characters vanish on G \ Z(G). In this paper, we find a necessary and sufficient condition for two VZ-groups to have identical character tables.

The following sections are included:

• Introduction

• Covering groups with proper subgroups

• The non-commuting graph

• The generating graph

• Solvable groups

• Artinian rings

• Arc transitive graphs

• Systems of distinct representatives

• Affine groups

• References

Let G = PSU(n,q) and let χ be a non-trivial unipotent character of G. In this paper, we study when the restriction of χ to H is reducible for any proper subgroup H of G, proving that this is always the case, apart from a few isolated exceptions.

In this note we extend [1, Theorem 3.10] to the context of compact groups. The invariance under weak forms of isoclinism will follow by some recent considerations in [2,3].

We try to answer the following question: Given a prime p and a subgroup H of group G, when do all the irreducible p-Brauer characters of G restrict irreducibly to H? We also address the same question for the irreducible π-partial characters of G.

In this paper several classical theorems on character degrees are revisited from the point of view of the real valued characters.

We study the following question. Let n be a positive integer and w a group-word. Consider the class of all groups G satisfying the identity wⁿ ≡ 1 and having the verbal subgroup w(G) locally finite. Is that a variety?

In the case that w = x the question is precisely the Restricted Burnside Problem. According to Zelmanov this has positive solution. We discuss results that show that the answer is positive for many other words w.

We describe the recent complete answer to the problem of the (2,3)-generation of the general and special linear groups over the integers.

Combining results of J-P. Serre and L. Ribes and P.A. Zalesskiĭ one concludes that a profinite group G acting without inversion of edges and with pro-p′ stabilizers on a pro-p tree T must be p-projective (Thm. B). It is shown that a finitely generated, p-projective virtual pro-p group G has such an action on a pro-p tree T (Thm. C). However, not every such profinite group G can act on a locally-finite pro-p tree without inversion of edges, with finite vertex stabilizers and with finitely many orbits (Thm. E). This fact is deduced from the existence of p-irrational, finitely generated, p-projective, virtual pro-p groups (Prop. D) using the theory of p-Lefschetz numbers.