Hello... I'm having difficulties solving the next problem:

Let G be a group, and N<G is anormalsub-group of G.

I'm given that |N|=n, and [G:N]=m.

a) prove that to every g in G, g^m is an elemnt in N.

I've proved that.

b) Now say that gcd(m,n) = 1. Prove that N is theonlysub-group of G with order n.

So, I'm having trouble with "b"... I don't seem to get anywhere.

And another question: I'm not given that G is finite, but is it not a consequence of "n" and "m" being finite? G is spanned by afinitenumber of cosets with afinitenumber of elements - I guess that means G cannot be infinite....

Thanks!!!