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A note on groups with the finite embedding property. (English) Zbl 0818.20028
Proceedings of the international conference on group theory, Timişoara, Romania, 17-20 September, 1992. Timişoara: Univ. Timişoara, Analele Universităţii din Timişoara. 43-45 (1993).
A multiplicative group $$G$$ is called an FE-group if, for any nonempty finite subset $$X$$ of $$G$$, there exists a finite group $$(H,*)$$ such that $$X\subseteq H$$ and $$xy= x*y$$ for all $$x,y\in X$$ with $$xy\in X$$. The reason of considering FE-groups is the following. Assume $$R=\bigoplus R_ g$$ ($$g\in G$$) is a $$G$$-graded ring with $$G$$ a group. Assume also that there exist only finitely many $$g\in G$$ such that $$R_ g$$ is nonzero. If this is the case, and if $$G$$ is an FE-group, then clearly $$R$$ becomes an $$H$$-graded ring with the same homogeneous components. Thus the theory of such a ring $$R$$ is in a sense reduced to the case $$G$$ is finite.
The author is interested in some properties of the class of FE-groups. Proposition 3 shows that FE-groups do not form a group variety: this class is closed under subgroups and direct products but it is not closed under factor groups. It has been proved [in S. Dăscălescu, C. Năstăsescu, A. del Rio, F. Van Oystaeyen, Gradings of finite support. Applications to injective objects (Preprint Univ. Antwerp, U.I.A. 1992)] that any locally residually finite group is an FE-group, and the paper ends with a question whether the converse is true (i.e. whether any FE-group is locally residually finite). The author himself thinks the answer to this question is no.
For the entire collection see [Zbl 0791.00021].
##### MSC:
 20E25 Local properties of groups 20E26 Residual properties and generalizations; residually finite groups 16W50 Graded rings and modules (associative rings and algebras) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth