On Schunck Classes of Finite Groups with Cover and Avoidance Property on Abelian Chief Factors

The main aim of this article is to obtain the characterization of local Schunck classes of finite groups whose projectors have the cover and avoidance property on Abelian chief factors.

Fitting sets pairs

On finite products of groups and supersolubility

Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Y(g) for some element g E G. i.e., XY(g) is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.

On conditional permutability and saturated formations

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ¿ ¿X, Y¿ for all X ¿ A and Y ¿ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups. © 2011 Edinburgh Mathematical Society.