Date: Mar 24, 2013 7:26 PM
Author: Jose Carlos Santos
Subject: Re: name for definition in group theory
On 24/03/2013 22:35, Arturo Magidin 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
Jose Carlos Santos