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The influence of $S$-quasinormality of some subgroups of prime power order on the structure of finite groups. (English) Zbl 1012.20009
A subgroup $H$ of the finite group $G$ is $S$-normal if it permutes with every Sylow subgroup of $G$. The author proves the following theorem. Let $G$ be a finite group. Assume $\pi(G)=\{p_1,\dots,p_n\}$ with $p_1>p_2>\cdots>p_n$ and $\exp\Omega(P_i)=p_i^{e_i}$ for $P_i\in\text{Syl}_{p_i}(G)$, $1\le i\le n$. Assume further that $\{H\mid H\le\Omega(P_i),\ H'=1,\ \exp H=p^{e_i}_i,\ 1\le i\le n\}$ consists of $S$-normal subgroups. Then $G$ is supersolvable. This theorem extends results of {\it M. Asaad, M. Ezzat} and the author [PU.M.A., Pure Math. Appl. 5, No. 3, 251-256 (1994; Zbl 0830.20034)] and the author [J. Egypt. Math. Soc. 5, No. 1, 1-7 (1997; Zbl 0915.20009)].

20D10Solvable finite groups, theory of formations etc.
20D20Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure
20D40Products of subgroups of finite groups