Alternating Group is Normal Subgroup of Symmetric Group

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Let $A_n$ be the alternating group on $n$ letters.

Then $A_n$ is a normal subgroup of $S_n$ whose index is $2$.

Proof
We have that $\operatorname{sgn} \left({S_n}\right)$ is onto $C_2$.

Thus from the First Isomorphism Theorem, $A_n$ consists of the set of even permutations of $S_n$.

The result follows from Subgroup of Index 2 is Normal.

Note
Note that when $n = 2$, we see immediately that $S_2 \cong C_2$ and the result still holds -- and here $A_n = \left\{{e_{S_2}}\right\}$.