On the second commutants of finite Alperin groups. (Q467651)

An Alperin group \(G\) is a group where every 2-generated subgroup has cyclic derived subgroup. Let \(p\) be an odd prime. Alperin first showed that finite Alperin \(p\)-groups are metabelian. This paper demonstrates that the structure of a general finite Alperin group is far from that of a finite Alperin \(p\)-group, in this sense: the author shows, by detailed computations, that for each finite abelian group \(H\) there exists a finite Alperin group \(G\) with \(G''\) isomorphic to \(H\). This furthermore holds when specified to finite Alperin 2-groups. The main theorem in this paper is the construction of a finite Alperin group \(G\) generated by \(n\geq 3\) involutions, with \(G''\) a direct product of copies of the cyclic group of order \(m\), where \(m\in\mathbb N\), the rank of \(G''\) is equal to \(\frac{(n-1)(n-2)}{2}\), and \(G''\) is contained in the centre of \(G\). A corollary of the results for finite groups, is that \(G\) is an Alperin group if and only if, for all elements \(a\) and \(b\) in \(G\), the commutators \([a,b,a]\) and \([a,b,b]\) are powers of \([a,b]\).