In a recent question of mine I asked whether every infinite group is (isomorphic to) the automorphism group of a graph. The finite case was done by Frucht in 1939. 


The first answer to this question pointed out two papers answering my original 
question, one by [Sabidussi](http://www.springerlink.com/content/qn32208134273016/)
and one by [de Groot](http://www.springerlink.com/content/w5446363757612w8/).


Reading the 3-page paper by Sabidussi I thought "Wow, these graphs are huge":
Sabidussi realizes a group of size $\kappa$ as the automorphism group of a graph of size $\aleph\_\kappa$. 


Indeed, de Groot in his paper notes that every countable group is the automorphism group
of a countable graph, every group of size $\leq 2^{\aleph\_0}$ is the automorphism group of a graph of size $\leq 2^{\aleph\_0}$, and every group of size $\kappa$ is the
the automorphism group of a graph of size $\leq 2^{\kappa}$.


But in general, he doesn't know how large a graph is needed to realize a given group.


Has this issue been resolved? Is there a reason why for a given infinite group $G$ there shouldn't be a graph of size $|G|$ whose automorphism group is isomorphic to $G$?


As I said in my original question, by Frucht's construction (and the constructions of de Groot and Sabidussi) this is related to the question whether there are $\kappa$ many non-isomorphic rigid graphs of size $\kappa$, where a graph is rigid if the identity is the only automorphism.  

Is this known? I would guess that there are $2^\kappa$ pairwise non-isomorphic rigid graphs of infinite size $\kappa$, but maybe I am wrong.



Groups as automorphism groups of small graphs and the number of rigid graphs of a given size
It is well-known that every infinite group $G$ can be realized as the automorphism group of a graph of size $|G|$. It is also well-known that for each infinite cardinal $\kappa$, there are $2^{\kappa}$ nonisomorphic rigid graphs of size $\kappa$. For example, both results are easily extracted from Section 4.2 of the following unpublished book:


<http://www.math.rutgers.edu/~sthomas/book.ps>


