Permutation of Cosets/Corollary 1
Jump to navigation
Jump to search
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 $9.23$