Stewart E. Stonehewer

The radicals of some group algebras

The Jacobson radicals of group algebras have received the attention of many authors over the years and several problems have remained unsolved. For example (see [#] and [.5]): If G is a group a& if F is a field of characteristic 0, is the group algebra FG semi-sinzple? (FG is semi-simple if its Jacobson radical J(FG) is triviahj It is known that FG is semi-simple in various special cases; for example, if

The subgroup lattice index problem

Abstract Given a group G and subgroups X ⩾ Y, with Y of finite index in X, then in general it is not possible to determine the index |X : Y| simply from the lattice of subgroups of G. For example, this is the case when G has prime order. The purpose of this work is twofold. First we show that in any group, if the indices |X : Y| are determined for all cyclic subgroups X, then they are determined for all subgroups X. Second we show that if G is a group with an ascending normal series with factors locally finite or abelian, and if the Hirsch length of G is at least 3, then all indices |X : Y| are determined.

Lattice homomorphisms of non-periodic groups

Usually we shall write simply t: C I + Y to denote the map r and speak of a complete /-homomorphism from G to 2’. We call T trivial if all subgroups of G have the same image under s; and we call T proper if r is not trivial and not injective. If (t) holds for all finite subsets .(/‘. then 7 is called a Irrtticr /2o111ot?1orpJli.o}?l (or I-i~~~~onlorpllisnz ). An /-homomorphism from G to the lattice /(G) of a group C? is called a projectiuit~~ if it is a bijection. Of course, a projectivity is always complete. A projectivity a: G --f G is said to be ilz~k~.~-prc.vercitl~ if. for K < H < G with 1 H : K1 finite.

On the Rarity of Quasinormal Subgroups

For each prime p and positive integer n, Berger and Gross have defined a finite p-group G = HX, where H is a core-free quasinormal subgroup of exponent p(n-1) and X is a cyclic subgroup of order p(n). These groups are universal in the sense that any other finite p-group, with a similar factorisation into subgroups with the same properties, embeds in G. In our search for quasinormal subgroups of finite p-groups, we have discovered that these groups G have remarkably few of them. Indeed when p is odd, those lying in H can have exponent only p, p(n-2) or p(n-1). Those of exponent p are nested and they all lie in each of those of exponent p(n-2) and p(n-1).

THE JOIN OF DISJOINT PERMUTABLE SUBGROUPS

ABSTRACT The authors obtain a bound for the derived length of a finite group, which is the join of two disjoint permutable nilpotent subgroups, in terms of the derived lengths of the two subgroups.

On groups with an interval isomorphic to a projective geometry

Abstract Let G be a group with a core-free subgroup H (≠1) such that the interval [ G / H ] is a projective geometry. Then H has a normal abelian complement (in G ) on which H acts faithfully and which is the direct product of H -isomorphic minimal normal subgroups of G provided either G is hyperabelian and H possesses a finite cyclic normal subgroup ≠1 (Theorem A) or G has an ascending normal series with finite abelian factors (Theorem B).

Generalising quasinormal subgroups

It has been shown by Cossey & Stonehewer (in the Padova Rendiconti 2011) that excessively large sections of finite p-groups can be devoid of quasinormal subgroups, bringing to an end the hope of relating group & subgroup-lattice structures via this concept. The present work begins the quest to find a generalisation of quasinormality that will be more successful. DOI: 10.1017/S0004972711003376

SOME FINITE SOLVABLE GROUPS WITH NON-TRIVIAL LATTICE ENDOMORPHISMS

The main purpose of this paper is to exhibit a doubly-infinite family of examples which are extensions of a p-group by a p′-group, with the action satisfying some conditions of Zappa (1951), arising from his study of dual-standard (meet-distributive) subgroups. The examples show that Zappas conditions do not bound the nilpotency class (or even the derived length) of the p-group. The key to this work is found in closely related conditions of Hartley (published here for the first time). The examples use some exceptional relationships between primes.