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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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