# Quotient of Symmetric Group by Alternating Group is Parity Group

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## 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 the quotient of $S_n$ by $A_n$ is the parity group $C_2$.

## Proof

From Alternating Group is Normal Subgroup of Symmetric Group, $A_n$ is a normal subgroup of $S_n$ whose index is $2$.

By definition of index, $\dfrac {S_n} {A_n}$ is a group of order $2$.

The result follows by definition of the parity group.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 81$