# 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 $\map \sgn {S_n}$ is onto $C_2$.

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

$\blacksquare$