Permutation of Cosets/Corollary 1

Theorem
Let $G$ be a group.

Let $H \le G$ such that $\index G H = n$ where $n \in \Z$.

Then:
 * $\exists N \lhd G: N \lhd H: n \divides \index G H \divides n!$

Proof
Apply Permutation of Cosets to $H$ and let $N = \map \ker \theta$.

Then:
 * $N \lhd G$ and $N \lhd H$

so from the Correspondence Theorem:
 * $H / N \le G / N$

such that:
 * $\index {G / N} {H / N} = n$

Thus:
 * $n \divides \index G N$

Also by Permutation of Cosets:
 * $\exists K \in S_n: G / N \cong K$

Thus:
 * $\index G N \divides n!$

as required.