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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
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