Regular orbits of cyclic subgroups in permutation representations of certain simple groups

Variants of some of the Brauer-Fowler theorems

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

On Thompson's Theorem

Finite Groups of Chain Difference One

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.

Given a set π of primes, say that a finite group G satisfies the Sylow π-theorem if every two maximal π-subgroups of G are conjugate; equivalently, the full analog of the Sylow theorem holds for π-subgroups. Say also that a finite group G satisfies the Baer-Suzuki π-theorem if every conjugacy class of G every pair of whose elements generate a π-subgroup itself generates a π-subgroup. In this article we prove, using the classification of finite simple groups, that if a finite group satisfies the Sylow π-theorem then it satisfies the Baer-Suzuki π-theorem as well.

Let x be a p-element of a finite group G. We say that x is unfused in G if, for some Sylow p-subgroup S of G containing x, all G-conjugates of x in S are S-conjugates. It is shown (using the classification of finite simple groups) that a finite group that contains an unfused involution has a chief factor of order 2. #Communicated by M. Dixon.

Let G be a finite group and p an odd prime. Let [Formula: see text] be the set of proper subgroups M of G with |G:M| not a prime power and |G:M|p=1. In this paper, we investigate the structure of G if every member of [Formula: see text] is nilpotent. In particular, a new characterization of PSL(2,7) is obtained. The proof of the theorem depends on the classification of finite simple groups.

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.

Introduction. In recent years, there have been interests in studying the influence of arithmetic conditions on the lengths of conjugacy classes to the structure of finite groups. The results are parallel to those of character degrees, and lead to strong restrictions to the group structure. An interesting relevant topic is the lengths of conjugacy classes of subgroups. This was first brought up to the author by Professor J. P. Zhang, who has obtained some interesting results on conjugacy class lengths of Sylow subgroups, the so called Sylow numbers [91. In this paper, we consider conjugacy class lengths of cyclic subgroups of finite goups. Our notations are standard and tends to follow [5]. All groups considered are finite. Let G be a finite group, H a subgroup of G, then the length of conjugacy class of H, the number of subgroups of G conjugate to H, is equal to I G : NG(H)[. Our main result is Theorem 1, its proof depends on the classification of finite simple groups.

Simple Connectedness of the 3-Local Geometry of the Monster

The purpose of this paper is to close one of the last gaps in the classification of finite simple groups containing a standard subgroup. We prove that a simple group containing a standard subgroup of type ${}^2{E_6}(2)$ has to be isomorphic to ${F_2}$, the baby monster.

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free. In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H1, …, Hm of some special form such that π(H1)∪…∪π(Hm) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections Hi so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H1,…,Hm} controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.