Let H be a subgroup of a group G. The permutizer \(P_G(H)\) is the subgroup generated by all cyclic subgroups of G which permute with H. A subgroup H of a group G is strongly permuteral in G if \(P_U(H)=U\) for every subgroup U of G, such that \(H\le U\le G\). We investigate groups with \(\mathbb {P}\)-subnormal or strongly permuteral Sylow subgroups. Moreover, we prove that groups with all strongly permuteral primary cyclic subgroups are supersoluble.

中文翻译：

---

具有置换主子群的有限群

设 H是群 G的子群。置换器\(P_G(H)\)是由G的所有循环子群通过H置换而生成的子群 。群 G的子群 H在G中是强置换的 ，如果对于G的 每个子群 U来说\(P_U(H)=U\)，使得 \(H\le U\le G\)。我们研究具有\(\mathbb {P}\)次正规或强置换 Sylow 子群的群。此外，我们证明了具有所有强置换主循环子群的群是超可溶的。