TITLE: Realizing groups as automorphism groups of graphs.
QUESTION [26 upvotes]: Frucht showed that every finite group is the automorphism group of a finite graph.  The paper is here.
The argument basically is that a group is the automorphism group of its (colored) Cayley graph
and that the colors of edge in the Cayley graph can be coded into an uncolored graph that has the same automorphism group.
This argument seems to carry over to the countably infinite case.
Does anybody know a reference for this?
In the uncountable, is it true that every group is the automorphism group of a graph?
(Reference?)
It seems like one has to code ordinals into rigid graphs in order to code the uncountably many colors of the Cayley graph.

REPLY [14 votes]: In the topological setting or if you want to relate the size of the graph to the size of the group, there are two relevant results:
(1) Any closed subgroup of $S_\infty$, i.e., of the group of all (not just finitary) permutations of $\mathbb N$, is topologically isomorphic to the automorphism group of a countable graph.
(2) The abstract group of  increasing homeomorphisms of $\mathbb R$, ${\rm Homeo}_+(\mathbb R)$, has no non-trivial actions on a set of size $<2^{\aleph_0}$. So in particular, it cannot be represented as the automorphism group of a graph with less than continuum many vertices.