On finite products of totally permutable groups

Abstract

In this paper the structure of finite groups which are the product of two totally permutable subgroups is studied. In fact we can obtain the ^-residual, where J-is a formation, Ti-projectors and %-normalisers, where 7i is a saturated formation, of the group from the corresponding subgroups of the factor subgroups.

On conditional permutability and factorized groups

“…. , G r in Theorem 1, Part 1 is true for any formation F containing U, and Part 2 holds if in addition F is either saturated or a formation of soluble groups ( [7], [8], [10]). …”

“…Motivated by the previous research on products of totally permutable subgroups and formations (see [7]- [10]) it is natural to ask whether those results can be achieved by weakening permutability to cc-permutability. A first approach to this study for products of tcc-permutable subgroups and saturated formations of soluble groups containing U was carried out in [3].…”

“…These factorized groups are said to be the product of the totally permutable subgroups A and B by R. Maier in [33]. Later on, a deep understanding of the structure of such groups has been reached and this study has been extended both in the frameworks of formation theory (see [7] - [10], [12], [15], [16], [33]) as well as in the theory of Fitting classes ( [22]- [24]). A detailed account on this topic can be found in the book [5].…”

Prefactorized subgroups in pairwise mutually permutable products

“…, G k if G i and G j are totally permutable subgroups of G, that is, every subgroup of G i permutes with every subgroup of G j for all i = j. These kind of products have been studied extensively with a lot of properties and results available (see [2,3,4,5,6,7,8,9, 10] and the papers cited therein).…”

Products of Pairwise Totally Permutable Groups

“…Moreover, the corresponding natural extensions for products of finitely many pairwise totally permutable groups also hold. We refer to [1][2][3] for details. For the dual type of classes, namely for Fitting classes containing U, the questions mentioned above were considered in [9].…”