1. ## automorphisms of Z_6

Hello

I'm having a problem with exercise II.4.8 of Hungerford's Algebra. It says: find an automorphism of $Z_6$ that is not an inner automorphism. An inner automorphism is one of the form $f(x)=gxg^{-1}$.

The problem is that every automorphism can be written as an inner automorphism because $Z_6$ is abelian.

Am I wrong?

(now i've got the latex)

2. ## Re: automorphisms of Z_6

Originally Posted by ModusPonens
Hello

I'm having a problem with exercise II.4.8 of Hungerford's Algebra. It says: find an automorphism of $Z_6$ that is not an inner automorphism. An inner automorphism is one of the form $f(x)=gxg^{-1}$.

The problem is that every automorphism can be written as an inner automorphism because $Z_6$ is abelian.
The fact that $Z_6$ is abelian does not imply that every automorphism is inner. What it does tell you is that the only inner automorphism is the identity map.

So any automorphism of $Z_6$ that is different from the identity must necessarily be outer. Can you find such a map? Hint: an automorphism of a cyclic group must take a generator of the group to a generator.

3. ## Re: automorphisms of Z_6

Of course! What was I thinking? Thank you.

4. ## Re: automorphisms of Z_6

Originally Posted by ModusPonens
Of course! What was I thinking? Thank you.
Just a remark, you always know precisely how big $\text{Inn}(G)$ is by the FIT since $\text{Inn}(G)\cong G/Z(G)$.