Frucht showed that every finite group is the automorphism group of a finite graph. The paper is [here](http://www.numdam.org/item?id=CM_1939__6__239_0).  

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.



Realizing groups as automorphism groups of graphs.
According to the [wikipedia page](http://en.wikipedia.org/wiki/Frucht%27s%5Ftheorem), *every* group is indeed the automorphism group of some graph. This was proven independently in 


de Groot, J. (1959), [*Groups represented by homeomorphism groups*](https://eudml.org/doc/160702), Mathematische Annalen 138 


and 


Sabidussi, Gert (1960), [*Graphs with given infinite group*](https://eudml.org/doc/177088), Monatshefte für Mathematik 64: 64–67.


