On conditional permutability and saturated formations

Abstract

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 Y g 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.

On conditional permutability and factorized groups

“…Also in [1] the behaviour of the supersoluble residual in products of finite groups is studied, by considering conditional permutability (not necessarily complete) as mentioned below in this Introduction (Theorem 2). Then, inspired by the previous research on totally permutable products of subgroups, an initial study on conditional permutability in the framework of formation theory has been developed in [3]. A compilation of recent results can be found in [2].…”

“…A compilation of recent results can be found in [2]. Easy examples in the previous references ([1, Examples 2,3]; also [3,Examples 3.5,3.6]) show that strong structural properties of products of totally permutable subgroups are missed when permutability is weakened to conditional permutability, even complete, and make evident the interest of the recent progress.…”

“…Example 3 in [1] (also [3,Example 3.6]) shows that this is not true if every subgroup of A is completely c-permutable with every subgroup of B, also it is not true for the supersoluble residuals. However, under this weaker hypothesis, we prove in this paper that the nilpotent residuals of the factors are normal subgroups in the product (Theorem 3).…”

“…More exactly, the supersoluble residuals of the factors centralize each other in a product of tcc-permutable subgroups. Moreover, Corollary 5 allows us to avoid restrictions to soluble groups in [3] and to extend the research in this reference to the universe of all finite groups (see Section 3).…”

“…Example 3 in [1] (also [3,Example 3.6]) shows that if the group G = AB is the product of tcc-permutable subgroups A and B, then A does not necessarily centralize B U .…”