A characterization of Schmidt groups by their endomorphism semigroups

Details A characterization of Schmidt groups by their endomorphism semigroups A characterization of Schmidt groups by their endomorphism semigroups

2002-10-24

arxiv.org/abs/math/0210380v1

Mathematics

Group Theory

LATEX2.09, 15 pages

Scientific paper

A {\it 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, $v\ge 1$. Denote by ${\mathcal{S}}$ the class of all Schmidt groups which have the same parameters p, q and v. It is shown in this paper that the class ${\mathcal{S}}$ can be characterized by the properties of the endomorphism semigroups of the groups of this class. It follows from this characterization that if $G\in {\mathcal{S}}$ and H is another group such that the endomorphism semigroups of G and H are isomorphic, then $H\in {\mathcal{S}}$, too.

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