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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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Jose Carlos Santos

Posts: 4,873
Registered: 12/4/04
Re: name for definition in group theory
Posted: Mar 24, 2013 7:26 PM
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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
>
> http://math.stackexchange.com/questions/28145/axiom-of-choice-and-automorphisms-of-vector-spaces/29469#29469


Thanks!

Best regards,

Jose Carlos Santos




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