# zbMATH — the first resource for mathematics

Influence of normality on maximal subgroups of Sylow subgroups of a finite group. (English) Zbl 0802.20019
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 $$G$$.
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 $$G/O_ p(G)$$ is supersolvable. In particular, $$G$$ is solvable.

##### MSC:
 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20E28 Maximal subgroups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D25 Special subgroups (Frattini, Fitting, etc.) 20D30 Series and lattices of subgroups
Full Text:
##### References:
 [1] B. Huppert, Endliche Gruppen 1 (Berlin–Heidelberg–New York, 1967). · Zbl 0217.07201 [2] D. Gorenstein, Finite Groups (New York, 1968). · Zbl 0185.05701 [3] J. S. Rose, A Course on Group Theory, Cambridge Univ. Press (London–New York–Melbourne, 1978). · Zbl 0371.20001 [4] R. Baer, Supersolvable immersion, Can. J. Math., 11 (1959), 353–369. · Zbl 0088.02402 [5] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Isr. J. Math., 35 (1980), 210–214. · Zbl 0437.20012 [6] W. R. Scott, Group Theory, Prentice–Hall (Englewood Cliffs, N. J., 1964).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.