# Subgroups and normal subgroups

1. Jan 19, 2009

### math8

Let G be a group, let N be normal in G and let H be a subgroup of G. Assume that G/N and H are finite and that gcd(|G/N|,|H|)=1. Prove that H is a subgroup of N.

I was thinking about using Lagrange Theorem. and maybe using the fact that G may act on the set of left cosets (G/N) by conjugation.
and the find the kernel of that action and then maybe use the first isomorphism theorem.

But I don't get very far with that.

2. Jan 19, 2009

### Dick

H also acts on G/N by left multiplication. Furthermore, every element of H is in one of the the left cosets of N.

3. Jan 19, 2009

### Hurkyl

Staff Emeritus
Maybe looking at G isn't the right way to go. What can you learn about H by studying the group G/N? What about N?