John Cossey

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

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.

ON PRODUCTS OF FINITE SUPERSOLUBLE GROUPS

In this paper two theorems of Kegel-Wielandt type are proved. As an application, we obtain sufficient conditions for m-permutable products of finite supersoluble groups to be supersoluble.

Sets of p-Powers as Conjugacy Class Sizes

We show that any finite set of powers of a fixed prime p which includes 1 can be the set of conjugacy class sizes of a p-group of nilpotency class 2. This corresponds to a result of Isaacs for degrees of irreducible characters.

Finite groups generated by subnormal T -subgroups

Our aim in this paper is to investigate the restrictions placed on the structure of a finite group if it can be generated by subnormal T -subgroups (a T -group is a group in which every subnormal subgroup is normal). For notational convenience we denote by the class of finite groups that can be generated by subnormal T -subgroups and by the subclass of of those finite groups generated by normal T -subgroups; and for the remainder of this paper we will only consider finite groups.

Classes of Finite Soluble Groups

I want to give here a rather biased account of recent work in the theory of classes of finite soluble groups. I will be concentrating on results which have something to say about the classes themselves, rather than results which use the classes to obtain a picture of the internal structure of finite soluble groups. My main excuse for doing so is that this part of the theory is at a very interesting stage: the classes are proving to be more exotic than might have been expected, and though we know little about them, some results and techniques are appearing, and it seems likely we will not remain so ignorant for long.

On the definition of saturated formations of groups

We exhibit a closure operation which serves to define saturated formations of finite soluble groups.

On some permutable products of supersoluble groups

It is well known that a group G = AB which is the product of two supersoluble subgroups A and B is not supersoluble in general. Under suitable permutability conditions on A and B, we show that for any minimal normal subgroup N both AN and BN are supersoluble. We then exploit this to establish some sufficient conditions for G to be supersoluble.

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 ).

On abnormal maximal subgroups of finite groups

An apparatus 115 for processing a substrate 20, comprises an integrated pumping system 155 having a high operating efficiency, small size, and low vibrational and noise levels. The apparatus 115 comprises a chamber, such as a load-lock chamber 110, transfer chamber 115, or process chamber 120. An integrated pump 165 is abutting or adjacent to one of the chambers 110, 115, 120 for evacuating gas from the chambers. In operation, the pump is located within the actual envelope or footprint of the apparatus and has an inlet 170 connected to a chamber 110, 115, 120, and an outlet 175 that exhausts the gas to atmospheric pressure. Preferably, the integrated pump 165 comprises a pre-vacuum pump or a low vacuum pump and is housed in a noise reducing enclosure having means for moving the pump between locations and means for stacking pumps vertically in use.