I have now thought about this and suspect that the answer to my own question is as follows:
For an automorphism we have
and from this it follows that
Am I correct?
Dummit and Foote Section 4.4 Automorphisms Exercise 1 reads as follows:
Let and is conjugation by g, prove that = .
A start to the proof is as follows:
Now we have completed the proof if
But why is in this case??
Can anyone please help clarify this situation?