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Zbl 1012.20009
Ramadan, M.
The influence of $S$-quasinormality of some subgroups of prime power order on the structure of finite groups. (English)
[J] Ann. Univ. Sci. Budap. Rolando E\H{o}tv\H{o}s, Sect. Math. 44, 3-9 (2001). ISSN 0524-9007

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)].
[U.Dempwolff (Kaiserslautern)]

MSC 2000:: *20D10 Solvable finite groups
20D20 Sylow subgroups of finite groups
20D40 Products of subgroups of finite groups

Keywords: $S$-quasinormal subgroups; Sylow subgroups; supersolvable finite groups

Citations: Zbl 0830.20034; Zbl 0915.20009

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