Some characterisations of \(\pi\)-solvable groups using index-complex (Q1611465)

Let \(G\) be a finite group, \(M\) be a maximal subgroup of \(G\) and \(H\) and \(K\) two normal subgroups of \(G\) such that \(H/K\) is a chief factor of \(G\). \(H\) is called a normal supplement of \(M\) in \(G\) if \(MH=G\). The normal index of \(M\) in \(G\) is defined as the order of \(H/K\) where \(H\) is minimal in the set of normal supplements of \(M\) in \(G\), and is denoted by \(\eta(G:M)\). It was proved that \(\eta(G:M)\) is uniquely determined by \(M\) [see \textit{W. E. Deskins}, Proc. Symp. Pure Math. 1, 100-104 (1959; Zbl 0096.24801)]. On this base, the authors generalize the Frattini subgroup and define the following characteristic subgroups: \(D_p(G)=\bigcap\{M\mid M\) is maximal and \([G:M]\) is composite and \(\eta(G:M)_p=1\}\), \(S_p(G)=\bigcap\{M\mid M\) is maximal and \(\eta(G:M)\) is composite and \(\eta(G:M)_p=1\}\), where \(p\) is a prime. With the help of these subgroups the authors obtain some results on \(\pi\)-solvable groups using the index-complex.