**Abstract:**

We introduce a variant of the notion of ‘locally embedable in finite groups’ (LEF) for metric groups, as ‘locally embedable in finite metric groups’ (LEFM). We show that every abelian group with an invariant metric is LEFM. We give a number of examples of classical abelian metric groups represented by nonstandard *finite abelian *metric groups using methods of nonstandard analysis. Generalising, we also define ‘locally embeddable into (uniformly) discrete metric groups’ (LEDM) and prove that every 2-sided invariant metric group is LEDM.

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School of Mathematics

University of Birmingham

http://web.mat.bham.ac.uk/R.W.Kaye/

Dr Walter Dean

Department of Philosophy

University of Warwick

http://go.warwick.ac.uk/whdean