On indices of subgroups in the join of their conjugate pairs. (Q376203)

All groups considered are finite. This paper analizes the influence of the index of \(H\) in \(\langle H,H^g\rangle\) for \(g\in G\) on the structure of a group \(G\), where \(H\) is either a second maximal subgroup of \(G\) or a Sylow subgroup of \(G\). The main results of the paper are the following: (Theorem 3.3) Let \(G\) be a group. Assume that \(|\langle L,L^g\rangle:L|\) is square-free for a second maximal subgroup \(L\) of \(G\) and \(g\in G\setminus N_G(L)\). Then either \(G\) is supersoluble, or \(G\) is an inner-supersoluble group (non-supersoluble group all of whose proper subgroups are supersoluble) with the following structure: \(G=RQ\), where \(R\) and \(Q\) are Sylow subgroups of \(G\), \(R\) is normal in \(G\) with \(\Phi(R)=1\), \(Q\) is cyclic, and \(\Phi(G)\leq\Phi(Q)\). (Theorem 4.3) Let \(G\) be a group. If \(|\langle P,P^g\rangle:P|\) is square-free for a Sylow subgroup \(P\) of \(G\) and \(g\in G\setminus N_G(P)\), then \(G\) is supersoluble.