On Graphs with Given Automorphism Group

Given that a large graph admits a group of automorphisms isomorphic to the abstract group G, what is the probability a(G) that G is its full automorphism group? This rational number can be computed from the structure of G. We determine the groups G with a(G) =1, show a(G) = 0 or 1 for all abelian groups G, and observe that the values of a(G) for metabelian groups G are dense in the unit interval.