Thread: 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 that is not an inner automorphism. An inner automorphism is one of the form .

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

Am I wrong?

(now i've got the latex)

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 that is not an inner automorphism. An inner automorphism is one of the form .

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

The fact that 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 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.

Of course! What was I thinking? Thank you.

Originally Posted by ModusPonens

Of course! What was I thinking? Thank you.

Just a remark, you always know precisely how big is by the FIT since .