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On products of nonnormal subgroups of finite soluble groups. (English) Zbl 1090.20009

Amberg, Höfling and Kazarin studied classes of finite groups closed under taking products of pairwise permutable subgroups. In a previous paper the author considered subgroup-closed formations closed under taking products of abnormal subgroups of soluble finite groups. In this article the author continues these investigations. The paper studies subgroup-closed saturated formations and Fitting formations closed under taking the products of nonnormal subgroups of finite soluble groups. A constructive description of soluble subgroup-closed saturated formations and Fitting formations of finite groups with the given property is obtained.
Recall that the normal closure of a subgroup \(U\) in a group \(G\) is the subgroup \(U^G=\langle U^g\mid g\in G\rangle\). The two main theorems of the paper give some equivalent statements to the statement: “If a group \(G=AB\) is a product of \(\mathcal F\)-subgroups \(A\) and \(B\) such that \(A^G=B^G=G\), then \(G\in{\mathcal F}\)” in the following two cases: 1) \(\mathcal F\) is a Fitting formation; 2) \(\mathcal F\) is a subgroup-closed saturated formation.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
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