Can a Group Have a Cyclic Automorphism Group of Odd Order?

[happyg1]

Homework Statement

Prove that no group can have its automorphism group cyclic of odd order.

Homework Equations

The Attempt at a Solution

Aut(Z2) has order 1, which is odd...trivial, yes, but I thought I was DONE.

However, my professor has said "well prove it EXCEPT for Z2"

I thought I was done, and now I have till 2 to do redo this and I have a mental block on it.

Can someone give my a push?

Thanks,
CC

 

[matt grime]

So you've got to show that every group (except Z/2Z) with cyclic aut. grp. has an automorphism of order 2...

 

[happyg1]

All right, I think I see it now...although I already turned it in incomplete. I just couldn't see it yesterday.
Thanks.