We exhibit a variation of the Lazard Elimination theorem for free restricted Lie algebras, and apply it to two problems about finite group actions on free Lie algebras over fields of positive characteristic.

The authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group.

Comments are made on the following question. Let m, n be positive integers and g a finite group. Suppose that for all choices of a subset of cardinality m and of a subset of cardinality n in g some member of the first commutes with some member of the second. Under what conditions on m, n is the group abelian?

Let k be a field of characteristic p > 0, G a finite p-solvable group and pm the highest power of p dividing the order of G. We denote by t(G) the nilpotency index of the (Jacobson) radical of the group algebra k[G]. The groups G with t(G) ≥ pm−1 are already classified. The aim of this paper is to classify the p-solvable groups G with pm−2 < t(G) < pm−1 for p odd.

The structure of finite groups in which permutability is transitive (PT-groups) is studied in detail. In particular a finite PT-group has simple chief factors and the p-chief factors fall into at most two isomorphism classes. The structure of finite T-groups, that is, groups in which normality is transitive, is also discussed, as is that of groups generated by subnormal or normal PT-subgroups.

A finite dinilpotent group G is one that can be written as the product of two finite nilpotent groups, A and B say. A finite dinilpotent group is always soluble. If A is abelian and B is metabelian, with |A| and|B| coprime, we show that a bound on the derived length given by Kazarin can be improved. We show that G has derived length at most 3 unless G contains a section with a well defined structure: in particular if G is of odd order, G has derived length at most 3.

We present some questions that we feel are important and interesting for the theory of finite p-groups, and survey known results regarding these questions.

Mike, the other year you asked me how I saw the future of p-group theory. All I can offer in answer are more questions.

We present a new algorithm which uses a cohomological approach to determine the groups of order pn, where p is a prime. We develop two methods to enumerate p-groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 29.

1991 Mathematics subject classification (Amer. Math. Soc.): primary 20D15.

Let r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

We survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of PI-algebras.

Suppose that G is a π-separable group. Let N be a normal π1-subgroup of G and let H be a Hall π-subgroup of G. In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G-conjugates ofHN. When N = 1 and π is the complement of a prime p, these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G.

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.