Finite groups with all subgroups not contained in the Frattini subgroup permutable. (Q652231)

A subgroup \(H\) of a group \(G\) is said to be `permutable' (or `quasinormal') if \(HK=KH\) for all subgroups \(K\) of \(G\). The structure of groups in which all subgroups are permutable (`quasihamiltonian groups') has been completely described by \textit{K. Iwasawa} [J. Fac. Sci. Univ. Tokyo, Sect. I 4, 171-199 (1941; Zbl 0061.02503)]. In the paper under review, the author proves that if \(G\) is a finite group in which all subgroups not contained in the Frattini subgroup \(\Phi(G)\) are permutable, then all subgroups of \(G\) are permutable. However, the proof presented is unnecessary, as the above statement can be obtained as a very easy consequence of previously known results in the following way. By hypothesis, the subgroup \(Q(G)\) generated by all non-permutable subgroups of \(G\) is contained in \(\Phi(G)\) and in particular \(Q(G)\neq G\). Then either all subgroups of \(G\) are permutable or the factor group \(G/Q(G)\) is cyclic [see \textit{M. Mainardis}, Rend. Semin. Mat. Univ. Padova 88, 245-261 (1992; Zbl 0782.20021)]; but in the latter case, also \(G/\Phi(G)\) is cyclic, so that \(G\) itself is cyclic and the statement is proved.