On the uniqueness of wreath products (Q1186765)

Let \(G=R\text{ wr }(P,I)\) be the restricted wreath product and \(M\) the base subgroup of \(G\). The author gives necessary and sufficient conditions for \(M\) to be characteristic subgroup in \(G\) when \(G\) is a finite group. More generally, if \(M\) is a normal subgroup of a finite group \(G\) where \(M\) is the direct product of isomorphic subgroups and \(G/M\) is faithfully represented as a permutation group acting by conjugation on the set of these subgroups, then it is shown that \(M\) is a characteristic subgroup of \(G\) unless very special conditions are satisfied.