Claim:

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:

Step 1:

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!).

Step 2:

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 .

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Done!