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On \(p\)-nilpotency of finite groups with some subgroups \(\pi\)-quasinormally embedded. (English) Zbl 1094.20007
A subgroup \(X\) of a finite group \(G\) is called \(\pi\)-quasinormal if \(XP=PX\) for each Sylow subgroup \(P\) of \(G\). Moreover, a subgroup \(H\) of \(G\) is said to be \(\pi\)-quasinormally embedded in \(G\) if for each prime divisor \(p\) of \(|H|\) any Sylow \(p\)-subgroup of \(H\) is also a Sylow \(p\)-subgroup of some \(\pi\)-quasinormal subgroup of \(G\). It has been proved by M. Asaad and A. A. Heliel [J. Pure Appl. Algebra 165, No. 2, 129-135 (2001; Zbl 1011.20019)] that if \(p\) is the smallest prime divisor of the order of a finite group \(G\), then \(G\) is \(p\)-nilpotent if and only if all maximal subgroups of Sylow \(p\)-subgroups of \(G\) are \(\pi\)-quasinormally embedded in \(G\).
In the paper under review the authors obtain some \(p\)-nilpotency criteria for finite groups related to the assumption that certain special subgroups are \(\pi\)-quasinormally embedded. For instance, they prove that if a finite group \(G\) contains a Sylow \(p\)-subgroup \(P\) such that \(N_G(P)\) is \(p\)-nilpotent and all maximal subgroups of \(P\) are \(\pi\)-quasinormally embedded in \(G\), then \(G\) itself is \(p\)-nilpotent.

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D40 Products of subgroups of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
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