Alternating Group is Normal Subgroup of Symmetric Group

Theorem
Let $n \ge 2$ be a natural number.

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 $\map \sgn {S_n}$ 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.

Examples
When $n = 2$, we have that:
 * $S_2 \cong C_2$

and:
 * $A_n = \set {e_{S_2} }$