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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 $\pi \left(G\right)=\left\{{p}_{1},\cdots ,{p}_{n}\right\}$ with ${p}_{1}>{p}_{2}>\cdots >{p}_{n}$ and $exp{\Omega }\left({P}_{i}\right)={p}_{i}^{{e}_{i}}$ for ${P}_{i}\in {\text{Syl}}_{{p}_{i}}\left(G\right)$, $1\le i\le n$. Assume further that $\left\{H\mid H\le {\Omega }\left({P}_{i}\right),\phantom{\rule{4pt}{0ex}}{H}^{\text{'}}=1,\phantom{\rule{4pt}{0ex}}expH={p}_{i}^{{e}_{i}},\phantom{\rule{4pt}{0ex}}1\le i\le n\right\}$ consists of $S$-normal subgroups. Then $G$ is supersolvable. This theorem extends results of 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)].
MSC:
 20D10 Solvable finite groups, theory of formations etc. 20D20 Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure 20D40 Products of subgroups of finite groups