CHARACTERIZATIONS OF SOME WREATH PRODUCTS OF GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

Abstract

Let A be a cyclic group of order pn, where p is a prime, and B be a finite abelian group or a finite p-group which is determined by its endomorphism semigroup in the class of all groups. It is proved that under these assumptions the wreath product A Wr B is determined by its endomorphism semigroup in the class of all groups. It is deduced from this result that if A, B, A0,…, An are finite abelian groups and A0,…, An are p-groups, p prime, then the wreath products A Wr B and An Wr (…( Wr (A1 Wr A0))…) are determined by their endomorphism semigroups in the class of all groups.