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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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Paul

Posts: 430
Registered: 7/12/10
Re: name for definition in group theory
Posted: Mar 24, 2013 5:34 PM
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On Sunday, March 24, 2013 9:03:01 PM UTC, Ken Pledger wrote:
> In article <m89uk8lshngii7afp7qktplg90ubnq9doj@4ax.com>,
>
> David C. Ullrich <ullrich@math.okstate.edu> wrote:
>
>
>

> > On Sun, 24 Mar 2013 08:15:15 -0700 (PDT), Paul <pepstein5@gmail.com>
>
> > wrote:
>
> >
>
> > >Does anyone know the name for the following property of a group G: G has
>
> > >no non-trivial automorphisms. ?
>
> > >Thank you
>
> >
>
> > These groups are referred to as "groups of order 1 or 2".
>
> >
>
> > There must be a very elementary proof of this. I know
>
> > no group theory; here's a not quite elementary proof
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> > using a big result from harmonic analysis....
>
>
>
>
>
> If G is non-Abelian, then it has non-trivial inner automorphisms
>
> x -> (a^(-1))xa. Therefore it's Abelian, so x -> x^(-1) is an
>
> automorphism, etc., as you showed.
>


Indeed, but this seems to be exactly what David said in his second posting. Of course, this doesn't quite complete the proof but David completed the argument. I find it non-obvious that the proof holds without AC, but I wasn't thinking about the axiom of choice until Jose raised it.

Paul Epstein




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