Cover-avoidance properties and the structure of finite groups. (English) Zbl 1028.20014

Let \(G\) be a finite group. The authors call a subgroup \(A\) of the group \(G\) a CAP-subgroup of \(G\) if for any chief factor \(H/K\) of \(G\) \(H\cap A=K\cap A\) or \(HA=KA\). Some characterizations for a finite group to be solvable are obtained when some of its maximal subgroups or \(2\)-maximal subgroups are CAP-subgroups. In particular, it is proved that a group \(G\) is solvable if and only if there exists a solvable \(2\)-maximal subgroup \(L\) of \(G\) such that \(L\) is a CAP-subgroup of \(G\).
It is also proved that \(G\) is \(p\)-solvable if and only if a Sylow \(p\)-subgroup \(P\) of \(G\) is a CAP-subgroup of \(G\).


20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D30 Series and lattices of subgroups
20E28 Maximal subgroups
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