On the Lattice of F-Dnormal Subgroups in Finite Soluble Groups

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.

Fitting classes and lattice formations I

AbstractA lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.