R. A. Bryce

The formation generated by a finite group

We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschutz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.

Subgroup coverings of some linear groups

A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is minimal if no other cover contains fewer members. We term minimised a minimal cover with the property that substituting for a member of the cover by a proper subgroup of that member produces a collection which is no longer a cover. We here describe the minimised covers for the groups GL2 (q), SL2(g), PSL2 (<?) and PGL2(9).

Covering groups with subgroups

A group is covered by a collection of subgroups if it is the union of the collection. The intersection of an irredundant cover of n subgroups is known to have index bounded by a function of n , though in general the precise bound is not known. Here we confirm a claim of Tompkinson that the correct bound is 16 when n is 5. The proof depends on determining all the ‘minimal’ groups with an irredundant cover of five maximal subgroups.

On certain abelian-by-nilpotent varieties

We show that, whenever m , n are coprime, each subvariety of the abelian-by-nilpotent variety has a finite basis for its laws. We further Show that the just non-Cross subvarieties of are precisely those already known.

Subgroup closed Fitting classes are formations

Since their introduction by Fischer(12) and Fischer, Gaschutz and Hartley (13) Fitting classes of soluble groups have attracted attention on two fronts (all groups considered in this paper will be finite and soluble). On the one hand is their important role in the structure of finite soluble groups, a good account of which can be found in Gaschutz (14), and on the other is their intrinsic interest as classes of groups. This paper falls into the second category, and is a continuation and completion of (8). There we proved that a subgroup closed Fitting class is a formation if it consists of groups of nilpotent length at most three. Happily, at last, we can remove this qualification.

A note on p -groups with power automorphisms

Let G be a group. The norm , or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G . In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω( G ). The Wielandt series of subgroups ω 1 ( G ) is defined by: ω 1 ( G ) = ω( G ) and for i ≥ 1, ω i+1 ( G )/ ω( G ) = ω( G /ω i , ( G )). The subgroups of the upper central series we denote by ζ i ( G ).

Bounds on the fitting length of finite soluble groups with supersoluble Sylow normalisers

We prove that, in a finite soluble group, all of whose Sylow normalisers are supersoluble, the Fitting length is at most 2m + 2, where p is the highest power of the smallest prime p dividing \G/G : here G is the supersoluble residual of G. The bound 2m + 2 is best possible. However under certain structural constraints on G/G , typical of the small examples one makes by way of experimentation, the bound is sharply reduced. More precisely let p be the smallest, and r the largest, prime dividing the order of a group G in the class under consideration. If a Sylow p-subgroup of G/G acts faithfully on every r-chief factor of G/G , then G has Fitting length at most 3.

A note on groups with non-central norm

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G . Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

Projective groups in varieties

We here give answers of sorts to a number of questions of Phi lip Ha I I. In a well-known paper [3] Ha I I defines the concept of splitting group in a variety of groups. (The reader is referred to [3], or to

A note on free products with a normal amalgamation

k of Chapter li of Hanna Neumanns book [4] for definitions.) In %h of that paper he finds various finite splitting groups in locally finite varieties. Since a group in a variety ¥ is a splitting group if and only if it is isomorphic to a complement of a normal subgroup in some free group of .V , the problem of finding splitting groups is in this sense the same as that of finding complemented normal subgroups in free groups of V. . Hence (paraphrasing (Ql) of [3]) one asks,