A CHARACTERIZATION OF SCHMIDT GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

Abstract

A Schmidt group is a non-nilpotent finite group in which each proper subgroup is nilpotent. Each Schmidt group G is a solvable group of order ps qv (where p and q are different primes) with a unique Sylow p-subgroup P and a cyclic Sylow q-subgroup Q, and hence G is a semidirect product of P by Q. Denote by [Formula: see text] the class of all Schmidt groups of orders ps qv, where p, q, and v are fixed and s is arbitrary. It is shown in this paper that the class [Formula: see text] can be characterized by the properties of the endomorphism semigroups of the groups of this class. It follows from this characterization that if [Formula: see text] and H is another group such that the endomorphism semigroups of G and H are isomorphic, then [Formula: see text], too.