On finite products of groups and supersolubility

Abstract

Two subgroups X and Y of a group G are said to be conditionallyi.e., X Y 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 factorized groups

“…Using these permutability properties new criteria for a product of finite supersoluble subgroups to be supersoluble are obtained in [21], [31], [32] and by the authors in [1], extending known results. 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).…”

“…On the other hand, we mention finally that for the particular case when F = U, the following result involving a weaker permutability condition was proved in [1,Theorem 2]. Theorem 2.…”

“…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].…”

“…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).…”

“…Example 3 in [1] (also [3, Example 3.6]) shows that this property fails when considering complete c-permutability instead of permutability. Nevertheless, we prove next that in a product of tcc-permutable subgroups each factor normalizes the nilpotent residual of the other factor.…”