A note on finite group structure influenced by second and third maximal subgroups (Q2640700)

The authors state (Theorem 2.1) that a finite group G which has a unique second maximal subgroup H (i.e., H is the unique maximal subgroup of every maximal subgroup of G) is necessarily a p-group. Since the symmetric group \(S_ 3\) is a counterexample with \(H=1\) it is apparently being assumed that H is nontrivial. In this case every maximal subgroup of G has a unique maximal subgroup, hence is cyclic, and the result follows easily. There are also some simple results for the cases when G has exactly two or exactly three second maximal subgroups but the presentation is somewhat confusing. The authors claim, for example, that a group with no second maximal subgroup can have order \(p^ 2\), p prime; it would seem that the trivial subgroup is satisfactory.