From: David C. Ullrich <ullrich@math.okstate.edu>
Subject: Re: Homomorphisms of a finite group
Date: Thu, 08 Feb 2001 17:02:21 GMT
Newsgroups: sci.math
Summary: Pontryagin duality: duals of locally compact abelian groups.

In article <95uh9l$tss$1@nnrp1.deja.com>,
  tim_brooks@my-deja.com wrote:
> Let G be a finite group and T the circle group, i.e the complex numbers
> z with |z|=1 under complex multiplication.
>
> what is known about hom(G,T), the Abelian group of group homomorphisms
> from G to T ?

If G is abelian then hom(G,T) is large enough to separate points of G.
In fact if G is abelian and just locally compact (possibly infinite)
then the _continuous_ members of hom(G,T) form a thing called the dual
group G*; turns out that G* also has a natural locally compact topology
and G**=G. This is the Pontryagin(?) duality theorem.

I happen to know that even though I know no algebra because it's
where abstract "harmonic analysis" starts. If G is not abelian I
imagine that in general hom(G,T) can be trivial - don't quote me
on that, though.

> Thanks,
> Tim
>
> Sent via Deja.com
> http://www.deja.com/
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