LAUSR

ABSTRACT Let be a cyclic group of order pm, where p is a prime and m is an arbitrary positive integer. It is proved that the wreath product (n factors) is determined for each natural n by its endomorphism semigroup in the class of all groups. It follows that (a) every Sylow subgroup of a finite symmetric group is determined by its endomorphism semigroup in the class of all groups, (b) each finite p-group G is embeddable into a finite p-group G* such that G* is determined by its endomorphism semigroup in the class of all groups.