Alan R. Camina

Sporadic groups and automorphisms of linear spaces

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let be a finite linear space on which G acts as a line-transitive automorphism group. Then the socle of G is not a sporadic group.

Block transitive automorphism groups of designs

In this note we consider subgroups of the automorphism group of a 2 - (v, k, 1) design. The notation is standard, see, for example, [2]. The number of blocks through a point is r = v - l/k - 1 and the number of blocks b is vr/k. Recall that a flag is an incident point-block pair. There are two main results in this note. The first is straightforward, but essential to the proof of the second.

The Wielandt length of finite groups

In [9] Wielandt introduced a subgroup of a group G defined as the intersection of the normalizers of all subnormal subgroups of G. I will call this subgroup the Wielandt subgroup E’(G). We can then define ?Vn(G) recursively by W,(G) = 1 and W,(G)/TJJ+,(G) = W[G/LV~/,,+,(G)]. If for some n, W,,(G) = : G and II is the least integer for which this happens, we say that G has Wielandt length n. LVielandt proved [9] that such an integer 11 always exists for finite groups. The main purpose of this paper is to investigate the relation between the Wielandt length of a group and various other invariants. The main result in this direction is Theorem I as follows:

The influence of conjugacy class sizes on the structure of finite groups: a survey

The importance of conjugacy classes for the structure of finite groups was recognised very early in the study of groups. In this survey we consider the results from the many articles which have developed this topic and examined the influence of conjugacy class sizes or the number of conjugacy classes on the structure of finite groups. Whilst we begin by mentioning the early results of Sylow and Burnside, our major objective is to highlight the more recent work and present some interesting questions which we hope will inspire further research.

A note on fitting classes

A subgroup V of G is called an j in jector of G if Vc~ N is F-maximal in N (that is, Vn N is a maximal ~-subgroup of N) for each subnormal subgroup N of G. It is known that the ~-injectors of a group G form a conjugacy class of subgroups 1,2]. If ~-injectors are always normal then ~ is called a normal Fitting class. In I-1] and [-3] some work has been done to classify normal Fitting classes. It has been shown in these papers that to any normal Fitting class there corresponds an object called a Fitting pair (f, A) defined as follows: A is an Abelian group and to each finite soluble group G there exists a homomorphism fG: G~A such that for all N ~ Gf~ = f~lN, where f~lN is the restriction of fG to N. Each such pair defines a Fitting class ~ given by G ~ if and only if fG(G)= 1, 1,Satz 3.1, 1.]. We use this to construct a Fitting class which contains

PRO-p GROUPS OF FINITE WIDTH

3, the symmetric group on 3 letters but does not contain the dihedral group of order 18. For some time it had been undecided whether the Fitting class generated by S 3 was the class consisting of all 3-groups extended by 2-groups. For a general survey see the paper of Cossey read at the conference on group theory held in Canberra, Australia 1973. The construction will actually provide uncountably many normal Fitting classes for all of which, the injectors have index 2. We will define and prove the existence of these classes by using Fitting pairs. Let f2 be the set of powers of odd primes and let 0 E f2. Let A be the cyclic group of order 2 and define, for any finite soluble group G,

Automorphisms of the Modular Curve

#The second author would like to acknowledge that a large amount of this research was carried out whilst she was employed by the University of the South Pacific.

COPRIME CONJUGACY CLASS SIZES

Let p>5 be a prime. Let X be the reduction of the modular curve X(p) in characteristic l (with l≠p). Aside from two known cases in characteristic l=3 (with p=7, 11), we show that the full automorphism group of X is PSL(2,p).

Intertwining automorphisms in finite incidence structures

We consider finite groups in which every triple of distinct conjugacy class sizes greater than one has a pair which is coprime. We prove such a group is soluble and has conjugate rank at most three.

Finite Soluble Hypernormalizing Groups

Abstract The automorphism group of a finite incidence structure acts as permutation groups on the points and on the blocks of the structure. We view these actions as linear representations and observe that they are intertwined by the incidence relation. Most commonly the intertwining is of maximal linear rank, so that the representation on points appears as a subrepresentation of the action of the blocks. The paper investigates various consequences of this fact.