# Math Help - normal subgroups and cayley's theorem

1. ## normal subgroups and cayley's theorem

Given H is a subgroup of G and that H has finite index n, prove that there is a normal subgroup K of G such that $K \leq H$ and $|G/K| \leq n!$. Don't assume that G is finite.

2. Note that $G$ acts on the cosets of $H$ by left multiplication. Since there are $n$ cosets, this induces a homomorphism $\phi: G\to S_n$. Then we know that $K=\ker \phi$ is a normal subgroup of $G$.

Try working with this.