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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 HG 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(π(G)) and assume that every maximal subgroup of the Sylow q-subgroups of G is normal in G for all qπ(G)-{p}. Then G has a Sylow tower and G/O p (G) is supersolvable. In particular, G is solvable.


MSC:
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-structure
20E28Maximal subgroups of groups
20D10Solvable finite groups, theory of formations etc.
20D25Special subgroups of finite groups
20D30Series and lattices of subgroups of finite groups
References:
[1]B. Huppert, Endliche Gruppen 1 (Berlin–Heidelberg–New York, 1967).
[2]D. Gorenstein, Finite Groups (New York, 1968).
[3]J. S. Rose, A Course on Group Theory, Cambridge Univ. Press (London–New York–Melbourne, 1978).
[4]R. Baer, Supersolvable immersion, Can. J. Math., 11 (1959), 353–369. · Zbl 0088.02402 · doi:10.4153/CJM-1959-036-2
[5]S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Isr. J. Math., 35 (1980), 210–214. · Zbl 0437.20012 · doi:10.1007/BF02761191
[6]W. R. Scott, Group Theory, Prentice–Hall (Englewood Cliffs, N. J., 1964).