This is a followup question to Pavel C's question here . It's fairly obvious from the axiom of choice that taking the direct sum of all finite groups produces the desired group. At the associated MathOverflow discussion referenced there, Joel David Hamkins noted there are countably infinitely many finite groups, so the resulting group is at most countably infinite. By Frucht's theorem, every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group G there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to G.
Here's my question: Is there an infinite undirected connected graph which corresponds to the group G constructed above where each of these non-isomorphic simple connected graphs are subgraphs of this graph-and if so, is this graph at most countably infinite or larger?
I think this is a great question with some deep implications for algebraic graph theory if it exists.
Of course,if a simple counterexample exists,I end up looking like a schmuck........lol