On the residual of a factorized group with widely supersoluble factors

Let \(\Bbb P\) be the set of all primes. A subgroup \(H\) of a group \(G\) is called {\it \(\mathbb P\)-subnormal} in \(G\), if either \(H=G\), or there exists a chain of subgroups \(H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \ \forall i.\) A group \(G\) is called {\it widely sup...

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Published in:arXiv.org 2020-03
Main Authors: Monakhov, Victor S, Trofimuk, Alexander A
Format: Article
Language:English
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Subgroups
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Summary:Let \(\Bbb P\) be the set of all primes. A subgroup \(H\) of a group \(G\) is called {\it \(\mathbb P\)-subnormal} in \(G\), if either \(H=G\), or there exists a chain of subgroups \(H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \ \forall i.\) A group \(G\) is called {\it widely supersoluble}, \(\mathrm{w}\)-supersoluble for short, if every Sylow subgroup of \(G\) is \(\mathbb P\)-subnormal in \(G\). A group \(G=AB\) with \(\mathbb P\)-subnormal \(\mathrm{w}\)-supersoluble subgroups \(A\) and \(B\) is studied. The structure of its \(\mathrm{w}\)-supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the \(\mathcal{A}\)-residual of \(G\). Here \(\mathcal{A}\) is the formation of all groups with abelian Sylow subgroups. Besides, we obtain new sufficient conditions for the \(\mathrm{w}\)-supersolubility of such group \(G\).
ISSN:2331-8422