Subgroup of Index 2 is Normal

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 $$g H = G - H$$ as there are only two cosets and the cosets partition $G$.

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

So every left coset is a right coset, and so by Normal Subgroup Equivalent Definitions: 1, $$H \triangleleft G$$.