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

 Messages: [ Previous | Next ]
 magidin@math.berkeley.edu Posts: 11,749 Registered: 12/4/04
Re: name for definition in group theory
Posted: Mar 24, 2013 6:35 PM

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

http://math.stackexchange.com/questions/28145/axiom-of-choice-and-automorphisms-of-vector-spaces/29469#29469

--
Arturo Magidin

Date Subject Author
3/24/13 Paul
3/24/13 David C. Ullrich
3/24/13 Paul
3/24/13 David C. Ullrich
3/24/13 Paul
3/24/13 Jose Carlos Santos
3/24/13 magidin@math.berkeley.edu
3/24/13 Jose Carlos Santos
3/24/13 Butch Malahide
3/24/13 Ken.Pledger@vuw.ac.nz
3/24/13 Paul
3/25/13 G. A. Edgar
3/25/13 G. A. Edgar
3/25/13 Paul
3/26/13 David C. Ullrich
3/25/13 David C. Ullrich