Another proof for the solvability of finite groups with at most two conjugacy classes of maximal subgroups (Q1071108)

It is an easy fact that a finite group having exactly one conjugacy class of maximal subgroups must be a cyclic group of prime power order (and consequently the group must have a unique maximal subgroup). The class of finite groups having exactly two conjugacy classes of maximal subgroups was studied up to some extend by \textit{A. Goncales} and \textit{C. Y. Ho} [Trabalho de Matemática No.163, Fund. Univ. Bras. (1980)] and \textit{S. Adnan} [Atti. Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 66, 175-178 (1979; Zbl 0448.20023)]. Later, it was shown by \textit{S. Adnan} [ibid. 68, 179 (1980; Zbl 0492.20015)] that such a group G must be of the form PQ where P and Q are Sylow subgroups of G with P normal in G, Q cyclic and Q acting irreducibly on P; in particular, G is solvable. The present author gives another (elementary) proof of solvability of such a group. \{Correction: On line 2 of the proof of Proposition 1 change \(\supsetneqq\) to \(\subsetneqq.\}\)