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The influence of $$s$$-conditional permutability of subgroups on the structure of finite groups. (English) Zbl 1240.20027
Summary: Let $$G$$ be a finite group. Fix a prime divisor $$p$$ of $$|G|$$ and a Sylow $$p$$-subgroup $$P$$ of $$G$$, let $$d$$ be the smallest generator number of $$P$$ and $$\mathcal M_d(P)$$ denote a family of maximal subgroups $$P_1,P_2,\dots,P_d$$ of $$P$$ satisfying $$\bigcap^d_{i=1}P_i=\Phi(P)$$, the Frattini subgroup of $$P$$. In this paper, we investigate the influence of $$s$$-conditional permutability of the members of some fixed $$\mathcal M_d(P)$$ on the structure of finite groups. Some new results are obtained and some known results are generalized.
##### MSC:
 20D40 Products of subgroups of abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D25 Special subgroups (Frattini, Fitting, etc.)
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