Let be a group and suppose there exists such that the map is an automorphism of Prove that for all
Note that is a homomorphism and so . Also, as it is onto every can be written in this form, for some .
Thus, it is sufficient to prove that .
Now, note that
.
Thus, as required.
I hope that's correct - haven't had a chance to check it properly as my wife is badgering me to go through and eat lunch!