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On the normal index and the \(c\)-section of maximal subgroups of a finite group. (English) Zbl 1203.20028
Let \(G\) a finite group and \(M\) a maximal subgroup of \(G\); the normal index of \(M\) in \(G\), denoted by \(\eta(G:M)\), is the order of a chief factor \(H/K\) such that \(H\) is a minimal supplement of \(M\) in \(G\). If \(H/K\) is such a chief factor then \(G=MH\), \(K\leq M\) and \((M\cap H)/K\) is called a \(c\)-section of \(M\). Finally if \(p\in\pi(G)\) then \(\mathcal F_p(G)\) is the set of maximal subgroups \(M\) of \(G\) such that \((|G:M|,p)=1\) and \(\mathcal F^p(G)\) is the set of maximal subgroups \(M\) of \(G\) such that \(N_G(P)\leq M\), where \(P\in\mathrm{Syl}_p(G)\).
Among other results the authors prove the following: Theorem 3.1. A group \(G\) is solvable if and only if \(\eta(G:M)_2=1\) for every \(M\in\mathcal F^2(G)\).
Theorem 3.4. A group \(G\) is solvable if and only if a \(c\)-section of \(M\) is either a \(2'\)-group or an Abelian \(2\)-group for every \(M\in\mathcal F^2(G)\).
Theorem 3.8. Let \(G\) be a group and \(p\in\pi(G)\). Then \(G\) is solvable if and only if \(\eta(G:M)\) is a power of a prime for every \(M\in\mathcal F_p(G)\).

20E28 Maximal subgroups
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
20D15 Finite nilpotent groups, \(p\)-groups
Full Text: DOI
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