On Sunday, March 24, 2013 12:10:01 PM UTC-5, José Carlos Santos wrote: > On 24-03-2013 16:44, David C. Ullrich wrote: > > > > > Since G is abelian, the map x -> -x is an automorphism. > > > Since this must be trivial, we have x + x = 0 for all > > > x. Hence G is a vector space over Z_2. And now as > > > above, if dim(G) = 0 or 1 then |G| = 1 or 2, while > > > if dim(G) > 1 then G has a non-trivial automorphism. > > > > Is this necessarily true without the axiom of choice?
No, it is not necessarily true without the Axiom of Choice. Without AC, one can construct a vector space over GF(2) that is nontrivial but has trivial automorphism group. See