# Finite group

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• Oct 7th 2011, 02:27 PM
dwsmith
Finite group
Let G be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g)=g$ iff. g = 1.

$(\Rightarrow)$
Suppose |G| = n and $\sigma(g)=g$.

First, I don't get why this would only work if g is 1.
• Oct 7th 2011, 02:30 PM
girdav
Re: Finite group
I don't understand what we have to show.
• Oct 7th 2011, 02:34 PM
dwsmith
Re: Finite group
Quote:

Originally Posted by girdav
I don't understand what we have to show.

Then you aren't alone.
• Oct 7th 2011, 02:41 PM
girdav
Re: Finite group
In your first post, you say that we have a finite group which possesses an automorphism $\sigma$ such that $\sigma (g)=g$ if and only if $g=1$.
Then, what do we have to do?
• Oct 7th 2011, 02:50 PM
dwsmith
Re: Finite group
Quote:

Originally Posted by girdav
In your first post, you say that we have a finite group which possesses an automorphism $\sigma$ such that $\sigma (g)=g$ if and only if $g=1$.
Then, what do we have to do?

Show it is a homomorphism, monic, and epi.
• Oct 7th 2011, 03:48 PM
Drexel28
Re: Finite group
Quote:

Originally Posted by dwsmith
Let G be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g)=g$ iff. g = 1.

$(\Rightarrow)$
Suppose |G| = n and $\sigma(g)=g$.

First, I don't get why this would only work if g is 1.

If I had to guess, you are asking about the common problem: "If $G$ is a finite group and $\sigma\in\text{Aut}(G)$ such that $\sigma^2=\text{id}$ and $\sigma$ possesses no non-identity fixed points, then $\sigma$ is the inverse map and $G$ is abelian."