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A Schmidt group is a non-nilpotent finite group in which each proper subgroup is nilpotent. Each Schmidt group G can be described by three parameters p, q and v, where p and q are different primes and v is a natural number, v ≥ 1. Denote by S the class of all Schmidt groups which have the same parameters p, q and v. It is shown in this paper that the class S can be characterized by the properties of the endomorphism semigroups of the groups of this class. It follows from this characterization that if G ∈ S and H is another group such that the endomorphism semigroups of G and H are isomorphic, then H ∈ S, too.

A CHARACTERIZATION OF SCHMIDT GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS