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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: 384
Registered: 7/12/10
Re: name for definition in group theory
Posted: Mar 24, 2013 12:43 PM
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On Sunday, March 24, 2013 4:25:18 PM UTC, David C. Ullrich 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
>
> using a big result from harmonic analysis:
>
>
>
> A topological group is a group together with a topology
>
> such that the group operations are continuous.
>
> A (continuous) character of a topological group G
>
> is a continuous homomorphism of G into the unit
>
> circle in the complex plane. If G is a topological
>
> group then the set of continuous characters is
>
> denoted G^*; note that G^* is itself a group,
>
> with multiplication defined pointwise.
>
>
>
> Now, if G is a locally compact abelian (LCA) group
>
> then the Pontryagin Duality Theorem states that
>
> G is isomorphic to its second dual (G^*)^*.
>
> That's the non-trivial part.
>
>
>
> (Oops, there's a missing definition there. If G is
>
> a LCA group then there is a natural topology on
>
> the group G^*; it turns out that G^* is also LCA.)
>
>
>
> Ok. Assume G is a group with no non-trivial
>
> automorphisms. Since all the inner automophisms
>
> of G are trivial, G must be abelian.
>
>
>
> Give G the discrete topology. Now G is an LCA
>
> group. Let K = G^*. (K is compact, not that
>
> we need that here.) Then G is isomorphic to
>
> K^*.
>
>
>
> Now, if chi is a character of K then chi^*, the
>
> complex conjugate, is also a character of K.
>
> The map chi -> chi^* is an automorphism of
>
> K^*. This automorphism must be trivial, so
>
> every chi in K^* must be real-valued.
>
>
>
> So every chi in K^* takes only the values 1 and -1.
>
> Hence every non-trivial element of K^* has order 2.
>
>
>
> So. G is an abelian group and every non-trivial element
>
> of G has order 2. This means that G is a vector space
>
> over the field Z_2 = {0,1}.
>
>
>
> If dim(G) = 0 or 1 then |G| = 1 or 2. If dim(G) > 1 then
>
> G has a non-trivial automorphism.
>
>
>
> The elementary proof would start by noting that G must
>
> be abelian, as above, and then invoke some structure
>
> theorem or something to deduce that every element
>
> has order 2, since any cyclic group of order greater
>
> than 2 has an automorphism...
>
>
>

> >
>
> >Paul Epstein

Thanks a lot. This raises a lot of interesting stuff and motivates me to research some of the underlying theory -- for example the proof of Pontryagin duality. I should have guessed (but failed to guess) that the reason there was no standard definition for the concept is that it's equivalent to something much simpler.
It's very impressive (to me) that you come up with these arguments so quickly.

Paul Epstein



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