  In this paper we formulate and study the problem of representing groups on
graphs. We show that with respect to polynomial time turing reducibility, both
abelian and solvable group representability are all equivalent to graph
isomorphism, even when the group is presented as a permutation group via
generators. On the other hand, the representability problem for general groups
on trees is equivalent to checking, given a group $G$ and $n$, whether a
nontrivial homomorphism from $G$ to $S_n$ exists. There does not seem to be a
polynomial time algorithm for this problem, in spite of the fact that tree
isomorphism has polynomial time algorithm.
