On groups of odd order and rank \(\leq 2\) (Q1079654)

Let G be a finite solvable group as in \textit{B. Huppert} [Endliche Gruppen I (1967; Zbl 0217.07201), p. 712]. Two theorems are proved. The first characterizes those finite groups G of odd order, with \(\Phi (G)=1\) which are of rank \(>2\) but all of their proper subgroups are of rank \(\leq 2\). These are semidirect products of an elementary abelian p-group V and a subgroup H of \(GL(V/F_ p)\), where H can be chosen in nine different ways, too complicated to be reproduced here. The second theorem gives necessary and sufficient conditions for an odd order group to be of rank \(\leq 2\).