Some Embedding Theorems for Groups
by
Peeter Puusemp
Department of Mathematics, Tallinn Technical University

Let G be a fixed group and End (G) the semigroup of all endomorphisms of G. If for an arbitrary group H the isomorphism of semigroups End (G) and End (H) implies the isomorphism of groups G and H, then we say that the group G is determined by its endomorphism semigroup (in the class of all groups). There are many groups that are determined by their endomorphism semigroups, for example, finite abelian groups, non-torsion divisible abelian groups finite symmetric groups, Sylow subgroups of finite symmetric groups. On the other hand, there exist also groups that are not determined by their endomorphism semigroups: the alternating group A₄, some Schmidt's groups, some semidirect products of finite cyclic groups.

In this connection, we have studied two following problems:

1) For a given group G find an extension K of G such that K is determined by its endomorphism semigroup in the class of all groups.

2) For a given group G find a group H such that the direct product G×H is determined by its endomorphism semigroup in the class of all groups.

In our talk we will characterize some results on these problems. Remark that similar problems for abelian groups and their endomorphism rings were studied by W.May [1].

Reference

1. May, W. Endomorphism rings of abelian groups with ample divisible subgroups. Bull. London Math. Soc., 1978, 10, no. 3, 270-272.

Date received: May 11, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-24.