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On pairwise mutually permutable products. (English) Zbl 1193.20020

A factorized group \(G=AB\) is the mutually permutable product of its subgroups \(A\) and \(B\) if \(A\) permutes with every subgroup of \(B\) and \(B\) with every subgroup of \(A\). The authors study the structure of finite groups \(G=G_1G_2\cdots G_n\) of finitely many pairwise mutually permutable subgroups \(G_i\) such that \(G_iG_j=G_jG_i\) for every pair \(i,j\).
It is shown that some relevant classes of groups are closed under forming pairwise permutable products. Among the classes considered are for instance the classes of \(p\)-supersoluble groups, the class of SC-groups (respectively SNAC-groups) in which every chief factor (every non-Abelian chief factor) is simple. If \(G\) is an SC-group, then necessarily so are all the factors \(G_i\) of \(G\), but this condition is sufficient only if the \(G_i\) belong to a special subclass of SC-groups.

MSC:

20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D30 Series and lattices of subgroups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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References:

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