Alternating Group is Set of Even Permutations

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

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

Then $A_n$ consists of the set of even permutations of $S_n$.

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 $S_n$.