Finite self dual groups.
A group \(G\) is said to be \(s\)-self dual if every subgroup of \(G\) is isomorphic to a quotient of \(G\). It is \(q\)-self dual is every quotient of \(G\) is isomorphic to a subgroup of \(G\). It is self dual if it is both \(s\)-self dual and \(q\)-self dual. Since a finite group is self dual \(\iff\) it is nilpotent and all its Sylow subgroups are self dual, it suffices to confine on prime power groups. Every Abelian group is selfdual. A nonabelian \(p\)-group \(G\) is \(s\)-self dual \(\iff\) either \(G=M\times E\), where \(M\) is of order \(p^3\) and exponent \(p\) and \(E\) is elementary Abelian, or \(G=A\times M\), where \(A\) is metacyclic minimal nonabelian of exponent \(p^n>p\) and \(M\) is Abelian of exponent \(<p^n\). The first series of groups and only these groups are self-dual \(p\)-groups.
scientific article; zbMATH DE number 6005425