Every group G is a subgroup of Isometry group of its Cayley graph.

What is essential property of being an Isometry group? Lie group?

  • $\begingroup$ What do you mean by asking what is an essential property of being a Lie group? Every group is a Lie group (just as every set is a zero-dimensional manifold). $\endgroup$ – David Carchedi May 11 '10 at 16:22
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    $\begingroup$ See also: mathoverflow.net/questions/993/… $\endgroup$ – Qiaochu Yuan May 11 '10 at 20:21

Every group is the full group of isometries of a connected, locally connected, complete metric space:

de Groot, J. "Groups represented by homeomorphism groups." Math. Ann. 138 (1959) 80–102. MR119193 doi:10.1007/BF01369667

Being a group of symmetries is the same thing as being a group.

You may also be interested to know that every group is the full automorphism group of a graph, not just a subgroup. References for this and various refinements are given at the wikipedia page for Frucht's theorem.

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    $\begingroup$ +1. I'd been wondering if Frucht's theorem had a name for about 10 years. $\endgroup$ – S. Carnahan May 11 '10 at 16:39

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