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.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Corollary $9.23$