Dualities of groups (Q678152)

Let \(G\) and \(\overline G\) be groups. A duality from \(G\) to \(\overline G\) is a bijection \(\delta\) from the set of all subgroups of \(G\) to the set of all subgroups of \(\overline G\) such that \(\delta\) and \(\delta^{-1}\) reverse inclusions. The Tarski groups are the first non-soluble examples of self-dual groups. The aim of this paper is to study the structure of groups admitting dualities. In particular modular groups with duals and locally graded groups admitting an autoduality satisfying some further requirements are considered.