Alternating Group - Knowlpedia

For $n\geq 2$ , the alternating group $A_n$ is the kernel

A_n=\ker\bigl(\operatorname{sgn}:S_n\to\{1,-1\}\bigr),

where $S_n$ is the group of permutations of $n$ elements under composition and $\operatorname{sgn}$ is the sign of a finite permutation.

Properties

Thus $A_n$ consists exactly of the even permutations.

The subgroup $A_n$ is normal in $S_n$ and has index $2$ , hence $|A_n|=n!/2$ . For $n\geq 5$ , $A_n$ is nonabelian simple.