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# alternating group is a normal subgroup of the symmetric group

###### Theorem 1.

The alternating group $A_{{n}}$ is a normal subgroup of the symmetric group $S_{{n}}$

###### Proof.

Define the epimorphism $f:S_{{n}}\rightarrow\mathbb{Z}_{2}$ by $:\sigma\mapsto 0$ if $\sigma$ is an even permutation and $:\sigma\mapsto 1$ if $\sigma$ is an odd permutation. Hence, $A_{{n}}$ is the kernel of $f$ and so it is a normal subgroup of the domain $S_{{n}}$. Furthermore $S_{{n}}/A_{{n}}\cong\mathbb{Z}_{2}$ by the first isomorphism theorem. So by Lagrange’s theorem

$|S_{{n}}|=|A_{{n}}||S_{{n}}/A_{{n}}|.$ |

Therefore, $|A_{{n}}|=n!/2$. That is, there are $n!/2$ many elements in $A_{{n}}$ ∎

Type of Math Object:

Theorem

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Reference

## Mathematics Subject Classification

20-00*no label found*

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