A note on the index complex of a maximal subgroup (Q1115963)

A subgroup \(C\) of finite group \(G\) is a completion of maximal subgroup \(M\) of \(G\) if \(C\not\subseteq M\) but every normal subgroup of \(G\) properly contained in \(C\) is in \(M\). The set of completions of \(M\) in \(G\) is the index complex of \(M\). It is proved here that \(G\) is solvable if (and only if) each of its maximal subgroups \(M\) avoids an abelian section derived from a maximal element of the index complex of \(M\). Also the maximal normal solvable subgroup of \(G\) is characterized as the intersection of those maximal subgroups which do not have avoiding abelian sections.