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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 π(G)={p 1 ,,p n } with p 1 >p 2 >>p n and expΩ(P i )=p i e i for P i Syl p i (G), 1in. Assume further that {HHΩ(P i ),H ' =1,expH=p i e i ,1in} 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:
20D10Solvable finite groups, theory of formations etc.
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-structure
20D40Products of subgroups of finite groups