The solubility graph associated with a finite group

The solubility graph associated with a finite group [Formula: see text] is a simple graph whose vertices are the elements of [Formula: see text], and there is an edge between two distinct elements [Formula: see text] and [Formula: see text] if and only if [Formula: see text] is a soluble subgroup of [Formula: see text]. We examine some properties of solubility graphs.

Finite Groups and Lie Rings with an Automorphism of Order 2n

Suppose that a finite groupGadmits an automorphismof order 2nsuch that the fixed-point subgroupof the involutionis nilpotent of classc. Letm=) be the number of fixed points of. It is proved thatGhas a characteristic soluble subgroup of derived length bounded in terms ofn,cwhose index is bounded in terms ofm,n,c. A similar result is also proved for Lie rings.

VARIATIONS SUR UN THÈME DE ALDAMA ET SHELAH

We consider a group G that does not have the independence property and study the definability of certain subgroups of G, using parameters from a fixed elementary extension G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

ON THE CENTRALIZER OF AN ELEMENT OF ORDER FOUR IN A LOCALLY FINITE GROUP

We prove that if G is a locally finite group admitting an automorphism φ of order four such that CG(φ) is Chernikov, then G has a soluble subgroup of finite index.

Some matrix groups over finite-dimensional division algebras

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.