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Zbl 0802.20019
Influence of normality on maximal subgroups of Sylow subgroups of a finite group.
(English)
[J] Acta Math. Hung. 59, No.1-2, 107-110 (1992). ISSN 0236-5294; ISSN 1588-2632/e

This paper deals with the influence of the normality of the maximal subgroups of Sylow $p$-subgroups of a finite group $G$ on the structure of $G$.\par Here is a sample of the results obtained: 1) If $G$ is a finite solvable group and every maximal subgroup of the Sylow subgroups of the Fitting subgroup $F(G)$ is normal in $G$, then $G$ is supersolvable. 2) Let $G$ be a finite group, let $H \triangleleft G$ and assume that $G/H$ is supersolvable and all maximal subgroups of the Sylow subgroups of $H$ are normal in $G$. Then $G$ is supersolvable. 3) Let $G$ be a finite group, let $p = \max(\pi(G))$ and assume that every maximal subgroup of the Sylow $q$-subgroups of $G$ is normal in $G$ for all $q \in \pi(G) - \{p\}$. Then $G$ has a Sylow tower and $G/O\sb p(G)$ is supersolvable. In particular, $G$ is solvable.
[M.Deaconescu (Safat)]
MSC 2000:
*20D20 Sylow subgroups of finite groups
20E28 Maximal subgroups of groups
20D10 Solvable finite groups
20D25 Special subgroups of finite groups
20D30 Series and lattices of subgroups of finite groups

Keywords: supersolvable groups; maximal subgroups; Sylow $p$-subgroups; finite group; finite solvable group; Fitting subgroup; Sylow tower

Cited in: Zbl 1080.20018 Zbl 1076.20010

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