A structure theorem is proved for finite groups with the property that, for some integer m with m ≥ 2, every proper quotient group can be generated by m elements but the group itself cannot.

In this paper a large family of dominant Fitting classes of finite soluble groups and the description of the corresponding injectors are obtained. Classical constructions of nilpotent and Lockett injectors as well as p-nilpotent injectors arise as particular cases.

Given a p-subgroup P of a finite group G we express the number of p-blocks of G with defect group P as the p-rank of a symmetric integer matrix indexed by the N(P)/P-conjugacy classes in PC(P)/P. We obtain a combinatorial criterion for P to be a defect group in G.

By a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

We solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.

The study of classes of finite groups is divided into two parts. The projective theory studies formations and Schunck classes. The dual injective theory studies Fitting classes. In each type of class a generalisation of Sylow's theorem holds. In this paper we seek further generalisations of Sylow's theorem which hold for classes which are neither injective nor projective, but obey other related properties. Firstly a common framework for the injective and projective theories is constructed. Within the context of this common framework further types of Sylow theorem can then be sought. An example is given of a property which is a simple hybrid of injectivity and projectivity which we will call ‘interjectivity’. A generalised Sylow theorem is then proved in the interjective case.

If G is a finite solvable group, we show that Isaacs' theory on partial characters on Hall π-subgroups can be developed for the nilpotent injectors of G. Therefore, the irreducible characters of G are partitioned into blocks associated to some nilpotent subgroups of G.

We show that p-groups of order p5 are determined by their group algebras over the field of p elements. Many cases have been dealt with in earlier work of ourselves and others. The only case whose details remain to be given here is that of groups of nilpotency class 3 for p odd.

A closure operation connected with Hall subgroups is introduced for classes of finite soluble groups, and it is shown that this operation can be used to give a criterion for membership of certain special Fitting classes, including the so-called ‘central-socle’ classes.

We give an improved bound for the order of a p-group enjoying the title property. We also point out relations between the upper central series of a p-group and of its maximal subgroups.

If R is a 2-group of symplectic type with exponent 4, then R is isomorphic to the extraspecial group , or to the central product 4 o 21+2n of a cyclic group of order 4 and an extraspecial group, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful representation of R of degree 2n. The 2-cocycle associated with this projective representation takes values which are powers of −1 if R is isomorphic to and powers of otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.

The main result of this paper is an upper bound for the number of maximal subgroups in finite solvable groups. Our result improves an earlier one of Cook, Wiegold and Williamson [1]. At the end, we use our bound to deduce an estimation for the total number of subgroups in finite solvable groups.

We consider (finite) groups in which every two-generator subgroup has cyclic commutator subgroup. Among other things, these groups are metabelian modulo their hypercentres, and in the corresponding quotient group all subgroups of the commutator subgroup are normal.

In a well-known paper, Hall and Higman proved the reduction theorem on a coprime order operator group acting on a finite group. This theorem plays an important role in local analysis of finite group theory. In this paper, we generalize the Hall-Higman reduction theorem by dropping the restrictive hypothesis (|G|, |H|) = 1 and determine the detailed structure of G completely.

If G is a π-separable group and χ is an irreducible character of G, then Issacs gas defined an associated pair (W, γ), called a nucleus of χ. The nucleus is the last term in a certain chain of pairs (I, Ω), where I is a subgroup of G and Ω is an irreducible character of I. The length of this chain is an invariant of χ that we call the nuclear length. In this paper we study bounds on the nuclear length of χ as a function of the π-length of G and as a function of the character degree χ(1).

There is a deeper structure to the ordinary character theory of finite solvable groups than might at first be apparent. Mauch of this structure, which has no analog for general finite gruops, becomes visible onyl when the character of solvable groups are viewes from the persepective of a particular set π of prime numbers. This purely expository paper discusses the foundations of this πtheory and a few of its applications. Included are the definitions and essential properties of Gajendragadkar's π-special characters and their connections with the irreducible πpartial characters and their associated Fong characters. Included among the consequences of the theory discussed here are applications to questions about the field generated by the values of a character, about extensions of characters of subgroups and about M-groups.

Coverings of certain simplical complexes constructed from subgroups of group G are related to covering groups of G.

We develop techniques to compute the homology of Quillen's complex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.

A practical method is described for deciding whether or not a finite-dimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalistaion of the Parker-Norton ‘Meataxe’ algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included.

Any representation of a group G on a vector space V extends uniquely to a representation of G on the free metabelian Lie algebra on V. In this paper we study such representations and make some group-theoretic applications.