Permutation of Cosets/Corollary 2
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Corollary to Permutation of Cosets
Let $G$ be a group.
Let $p$ be the smallest prime such that:
- $p \divides \order G$
where $\divides$ denotes divisibility.
Let $\exists H: H \le G$ such that $\order H = p$.
Then $H$ is a normal subgroup of $G$.
Proof
Apply Permutation of Cosets: Corollary 1 to $H$ to find some $N \lhd G$ such that:
- $\index G N \divides p!$
Because $\index G N \divides \order G$, it divides $\gcd \set {\order G, p!}$.
Because $p$ is the smallest prime dividing $\order G$, it follows that:
- $\gcd \set {\order G, p!} = p$
Thus:
- $\index G N = p = \index G H$
Because $N \subseteq H$, it must follow that $N = H$.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Corollary $9.25$