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


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



What kind group can be realized as a Isometry group of some space?
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](http://www.ams.org/mathscinet-getitem?mr=119193)
[doi:10.1007/BF01369667](http://dx.doi.org/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](http://en.wikipedia.org/wiki/Frucht%27s_theorem).


