Hi everyone!
I have the next problem:
a finit group which has order and defined by , I need to find a necessary and sufficient condition so that is an isomorphism.
Any suggestion?
Thanks in advance!
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 .
__________________________________________________ _____________________________
Done!
Thanks for your answer! I forgot to said that was abelian but you know this. I don't understand in the exercise 4 how using that you can prove and if you prove this you have that is an isomorphism? or you have to prove that is surjective too?
Thanks again!
for maps on finite sets, injective implies surjective and vice versa.
now think about what it means for g to be in ker(φ). it means .
there are two ways this could happen: g = e, or |g| divides m. suppose that |g| divides m.
since |g| also divides |G| = n, |g| is a common divisor, so.....