
Re: name for definition in group theory
Posted:
Mar 24, 2013 12:25 PM


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 nontrivial 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 nontrivial 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 nontrivial 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 realvalued.
So every chi in K^* takes only the values 1 and 1. Hence every nontrivial element of K^* has order 2.
So. G is an abelian group and every nontrivial 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 nontrivial 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

