Hello... I'm having difficulties solving the next problem:
Let G be a group, and N<G is a normal sub-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 the only sub-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 a finite number of cosets with a finite number of elements - I guess that means G cannot be infinite....