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Topic: name for definition in group theory
Replies: 15   Last Post: Mar 26, 2013 11:35 AM

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Posts: 11,749
Registered: 12/4/04
Re: name for definition in group theory
Posted: Mar 24, 2013 6:35 PM
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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

Arturo Magidin

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