L. G. Kovács

On finite soluble groups

This is best possible in the sense that to each positive value of d one can easily construct such groups G (in fact, abelian-by-cyclic groups) which cannot be generated by d elements. A suitable rearrangement of the steps of GASCrlt)TZ [2] can give further results such as the following: if every abelian-bycyclic factor group of G can be generated by k elements and k> 89 (d+ 1), then G can be generated by k elements. The author is indebted to Dr. M. F. N~WMAN for the discussions which led to this paper. It is convenient to put the first steps of the proof of Theorem 1 separately:

On minimal faithful permutation representations of finite groups

The minimal (faithful) degree p(G) of a finite group G is the least positive integer n such that G < Sn. Clearly if H ^ G then n{H) < n(G). However if N < G then it is possible for fi(G/N) to be greater than fi(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.

On critical groups

The concept of critical group was introduced by D. C. Cross (as reported by G . Higman in [5]): a finite group is called critical if it is not contained in the variety generated by its proper factors. (The factors of a group G are the groups H/K where K H ≦ G , and H/K is a proper factor of G unless H = G and K =1). Some investigations concerning finite groups and varieties depend on the investigators ability to decide whether a given group is critical or not. (For instance, one of the crucial points in the important paper [9] of Sheila Oates and M. B. Powell is a necessary condition of criticality: their Lemma 2.4.2.) An obvious necessary condition is that the group should have only one minimal normal subgroup: the group is then called monolithic , and the minimal normal subgroup its monolith . This is, however, far from being a sufficient condition, and it is the purpose of the present paper to give some sufficient conditions for the criticality of monolithic groups. (We consider the trivial group neither monolithic nor critical.) The basis of our results is an analysis of the following situation.

On normal subgroups which are direct products

Now that the classification of finite simple groups is complete, it is logical to look at the extension problem. An important special case to consider is when M is a minimal normal subgroup of G and both G/M and M are known groups. If M is abelian, various techniques have been used to derive information about G. Indeed, almost the entire theory of finite solvable groups can be said to rest upon these techniques. The motivation behind the present paper was to develop techniques for dealing with the situation when M is not abelian. Specifically, we consider the following problems: (1) Determine the structure of G from the structure of G/M and some subgroup or subgroups of G. (2) Find subgroups H in G such that G = HM, and, in particular, determine whether M has a complement in G. (3) Determine when two subgroups H, and H, found in (2) are conjugate in G. If M is a non-abelian minimal normal subgroup of a finite group G, then

The presentation rank of a direct product of finite groups

where I’ is EG-projective and A has no projective direct summand. By a theorem of Swan [7],1’ @ Q E (QG)s (th e d. erect sum of s topics of QG) for some non-negative integers. It is known that this integer is independent of the particular minimal free presentation and the particular decomposition of R (cf. [3, pp. 263-2641). It is therefore an invariant of G that we call the presentation rank of G and write pr(G) s. It has been shown elscwhcre that all 2-generator groups (and therefore all known simple groups) as well as all soluble groups have zero presentation rank ([3, p. 2671 and [4], respectively). This might suggest that there do not exist any groups with non-zero presentation rank. Our aim here is to show that this is far from being the case by establishing

Cross varieties of groups

One of the fundamental problems in the theory of varieties of groups is to decide whether the laws (identical relations) of each group admit a finite basis (in the sense that they are all consequences of a finite set of laws). Oates & Powell recently proved (1964) that the answer is affirmative in the case of finite groups. We present a considerably shortened proof of their result; with a little additional reasoning, this in fact yields a slight generalization of the Oates-Powell theorem.

Maximal subgroups in composite finite groups

The purpose of this paper is to present a method for translating the problem of finding all maximal subgroups of finite groups into questions concerning groups that are nearly simple. (A finite group is called nearly simple if it has only one minimal normal subgroup and that it nonabelian and simple.) In view of the recently announced classification of all finite simple groups this seems to be a useful reduction, though it must be acknowledged (see, e.g., Scott [6]) that there are still enormous obstacles on the way to understanding even just the maximal subgroups of the simple groups. Let G be a finite group and M a minimal normal subgroup of G. The maximal subgroups of G containing M are of course in bijective correspondence with the maximal subgroups of the smaller group G/M, and so we need not concern ourselves with those. If M is abelian, the maximal subgroups of G not containing M are precisely the complements of M in G; the number of conjugacy classes of these is 0 or the order of the first cohomology group ~‘(G/~, M). The case of A4 nona~lian and nonsimple is the principal part of this paper. If neither reduction is applicable, then all minimal normal subgroups of G are nonabelian simple groups: this is dealt with in the entirely straightforward penultimate section of the paper. These reductions are all “canonical” or “natural” in a sense which could perhaps be expressed in the language of categories and functors, but here we prefer to stay with older conventions. (In particular, we usually do not distinguish between a homomorphism and that obtained from it by restricting the codomain.) Nevertheless, the interested reader will observe that much of the strength of the results lies precisely in their canonical nature, implicit as it may remain in this exposition. While I do not know of any explicit statement of this reduction 114 002 l-8693/86 IE3.00

On non-cross varieties of groups

Our title has become something of a misnomer, however we retain it since drafts of this note have been quoted with it. Unless otherwise stated our terminology and notation follow that in Hanna Neumanns book [12]. The Oates-Powell Theorem ([12] p. 151) allows us to say that a variety is Cross if and only if it can be generated by a finite group, and to assert that the laws of a Cross variety are finitely based. A variety is just-non-Cross if it is not Cross but every proper subvariety of it is Cross. We asked in [9]: what non-Cross varieties have just-non-Cross subvarieties? The answer is: all of them.

Symmetric units in modular group algebras

Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g\mapsto g\m1 of G extends linearly to an anti-automorphism a\mapsto a^* of KG. An element a of KG is called symmetric if a^*=a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group.

Unitary units in modular group algebras

AbstractLetp be a prime,K a field of characteristicp, G a locally finitep-group,KG the group algebra, andV the group of the units ofKG with augmentation 1. The anti-automorphismg→g−1 ofG extends linearly toKG; this extension leavesV setwise invariant, and its restriction toV followed byv→v−1 gives an automorphism ofV. The elements ofV fixed by this automorphism are calledunitary; they form a subgroup. Our first theorem describes theK andG for which this subgroup is normal inV.For each elementg inG, let