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On complemented subgroups of finite groups. (English) Zbl 0929.20015

Let \(G\) be a finite group. If \(N\) is a normal subgroup of \(G\) such that \(G/N\) is supersoluble and every minimal subgroup of \(N\) is complemented in \(G\), then \(G\) is supersoluble (Theorem 1). If all maximal subgroups of every Sylow subgroup of \(G\) are complemented in \(G\), then \(G\) is supersoluble (Theorem 2). If \(p\) is the smallest prime dividing the order of \(G\), all 2-maximal subgroups of every Sylow \(p\)-subgroup of \(G\) are complemented in \(G\) and \(G\) is \(A_4\)-free, then \(G\) is \(p\)-nilpotent (Theorem 3). If \(G\) has even order and \(K\) is a maximal subgroup of a Sylow-2 subgroup of \(G\) such that \(K\) is complemented in \(G\), then \(G\) is not nonabelian simple and a composition factor of \(G\) is either a cyclic group of prime order or \(B=\text{PSL}(2,q)\) for some Mersenne prime \(q\) such that a Hall \(2'\)-subgroup of \(B\) is the normaliser of a Sylow \(q\)-subgroup of \(B\) (Theorem 4).

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D40 Products of subgroups of abstract finite groups
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