Permutation of Cosets/Corollary 1

Theorem
Let $G$ be a group.

Let $H \le G$ such that $\left[{G : H}\right] = n$ where $n \in \Z$.

Then:
 * $\exists N \lhd G: N \lhd H: n \mathop \backslash \left[{G : N}\right] \mathop \backslash n!$

Proof
Apply Permutation of Cosets to $H$ and let $N = \ker \left({\theta}\right)$.

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

so $H / N \le G / N$ such that $\left[{G / N : H / N}\right] = n$ from the Correspondence Theorem.

Thus $n \mathop \backslash \left[{G : N}\right]$.

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

Thus $\left[{G : N}\right] \mathop \backslash n!$ as required.