# Subgroup of Index 2 is Normal

From ProofWiki

## Theorem

A subgroup of index 2 is always normal.

## Proof

Suppose $H \le G$ such that $\left[{G : H}\right] = 2$.

Thus $H$ has two left cosets (and two right cosets) in $G$.

If $g \in H$, then $g H = H = H g$.

If $g \notin H$, then $g H = G \setminus H$ as there are only two cosets and the cosets partition $G$.

For the same reason, $g \notin H \implies H g = G \setminus H$.

That is, $g H = H g$.

The result follows from the definition of normal subgroup.

$\blacksquare$

## Sources

- Richard A. Dean:
*Elements of Abstract Algebra*(1966)... (previous)... (next): $\S 1.10$: Theorem $23$ - Allan Clark:
*Elements of Abstract Algebra*(1971)... (previous)... (next): $\S 46 \gamma$ - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): $\S 49.2$ - John F. Humphreys:
*A Course in Group Theory*(1996)... (previous)... (next): $\S 7$: Example $7.6$