Products of locally cyclic groups

We consider groups of the form $${G} = {AB}$$ G = AB with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups. As an application, it is shown that the Prüfer rank of a periodic product of two locally cyclic subgroups does not exceed 3, and this bound is sharp. It is also proved that a product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group. This generalizes a well-known theorem of B. Huppert for finite groups.

Nonunits of group algebras over the fours group

The conjecture on units of group algebras of a torsion-free supersoluble group is saying that every unit is trivial, i.e. a product of a nonzero element of the field and an element of the group. This conjecture is still open and even in the slightly simple case of the fours group [Formula: see text], it is not yet known. The main result of this paper is to show that a wide range of elements of group algebra of [Formula: see text] are nonunit.

Probability and Bias in Generating Supersoluble Groups

We discuss some questions related to the generation of supersoluble groups. First we prove that the number of elements needed to generate a finite supersoluble groupGwith good probability can be quite a lot larger than the smallest cardinality d(G) of a generating set ofG. Indeed, ifGis the free prosupersoluble group of rankd⩾ 2 and dP(G) is the minimum integerksuch that the probability of generatingGwithkelements is positive, then dP(G) = 2d+ 1. In contrast to this, ifk–d(G) ⩾ 3, then the distribution of the first component in ak-tuple chosen uniformly in the set of all thek-tuples generatingGis not too far from the uniform distribution.

Ensuring a finite group is supersoluble

A special case of the main result is the following. Let G be a finite, non-supersoluble group in which from arbitrary subsets X, Y of cardinality n we can always find x ∈ X and y ∈ Y generating a supersoluble subgroup. Then the order of G is bounded by a function of n. This result is a finite version of one line of development of B.H. Neumann's well-known and much generalised result of 1976 on infinite groups.

On the Wielandt length of a finite supersoluble group

We develop a technique for calculating the Wielandt subgroup of a semidirect product of two finite groups of coprime order. We apply this technique to calculate the Wielandt length of a supersoluble group in terms of the Wielandt lengths of its Sylow subgroups (for small Wielandt lengths) and in terms of the nilpotency classes of its Sylow subgroups.

A conjecture of Lennox and Wiegold concerning supersoluble groups

We prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

Groups sharing some varietal properties with supersoluble groups

In this note a formation U is considered which can be defined by a sequence of laws which ‘almost’ hold in every finite supersoluble group. The class U contains all finite supersoluble groups and each group in U has a Sylow tower.It is shown that a finite group belongs to U if and only if all of its subgroups with nilpotent commutator subgroup are supersoluble. A more general result concerning classes of this type finally proves that U is a saturated formation.

Zero divisors and idempotents in group rings

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.