The aim of this paper is to prove the following result: Let π be a set of odd primes. If the finite group G = AB is a product of two π-decomposable subgroups A = Oπ(A) × O π (A) and B = Oπ(B) × O π (B), then Oπ(A)Oπ(B) = Oπ(B)Oπ(A) and this is a Hall π-subgroup of G.

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

A bstract Let G be a finite group and p a prime. We consider an 1injecta r K of G, being 1 a Fitting class between~p* p and~P* ep, and we study the structure and normality in G of the subgroups ZJ(K) and ZJ* (K), provided that G verify certain conditions, extendin g some results of G. Glauberman (A characteristic subgroup of a pstable group, Canad.

No abstract