defined by is an isomorphism iff G is abelian and
will stand for composed with itself 'p' times. (Naturally )
Exercise 1: Prove that if is an isomorphism then is an isomorphism too.
Assume is an isomorphism and you can prove the conditions in two steps:
Let . Then write ma + nb = d for some integers a and b (Why can you do this?).
[Can I do the above step if 'a' is negative?]
Exercise 2:What happens, if ?
[Hint: Look at and rule out this case]
If d=1, we have and,
Exercise 3: is a homomorphism iff G is abelian (Prove it!).
Exercise 4:Verify: If G is abelian and , then is an isomorphism.
This part is easy. Use the abelian nature to prove the function is a homomorphism and use to prove .