The structure of 3-pyramidal groups (Q6054763)

A finite group \(G\) is 3-pyramidal if it has exactly 3 involutions, which are all conjugate to each other. The main result proved in the paper under review is (see Theorem 1.2): Let \(G\) be a finite group, \(O(G)\) the largest normal subgroup of \(G\) of odd order and let \(K\) be the subgroup generated by the involutions of \(G\). Then \(G\) is 3-pyramidal if and only if one of the following holds. \begin{itemize} \item[(1)] \(G \simeq S_{3} \times H\) where \(H\) is a group of odd order. \item[(2)] \(O(G) \leq C_{G}(K)\) and \(G/O(G) \simeq N \rtimes A\) where \(N\) is the Suzuki 2-group of order 64 and \(A\) is cyclic of order 3 or 15. \item[(3)] \(O(G) \leq C_{G}(K)\) and \(G/O(G)\) is isomorphic to \((C_{2^{n}} \times C_{2^{n}}) \rtimes A\) where \(A\) is the cyclic group of order 3 acting fixed point freely on \(C_{2^{n}} \times C_{2^{n}}\). \end{itemize}