Clones of finite groups. (Q2577746)

Let \(G\) and \(H\) be groups such that the subgroup lattices of \(G^3=G\times G\times G\) and \(H^3=H\times H\times H\) are isomorphic. The question whether then \(G\) and \(H\) are isomorphic has been answered, in the negative, by the authors [J. Group Theory 7, No. 3, 385-402 (2004; Zbl 1071.20028)]. If \(G\) and \(H\) are finite and have Abelian Sylow subgroups, more can be said: for each \(\kappa\), the subgroup lattices of \(G^\kappa\) and \(H^\kappa\) are isomorphic. This is not enough to imply that \(G\) and \(H\) are isomorphic.