
Re: name for definition in group theory
Posted:
Mar 24, 2013 6:35 PM


On Sunday, March 24, 2013 12:10:01 PM UTC5, José Carlos Santos wrote: > On 24032013 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 nontrivial 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
http://math.stackexchange.com/questions/28145/axiomofchoiceandautomorphismsofvectorspaces/29469#29469
 Arturo Magidin

