I know how to prove that $A_n$ is a subgroup of $S_n$ but i do not know how to prove this using the index.
2026-03-30 04:59:07.1774846747
Prove that $A_n$ is a Normal subgroup of $S_n$?
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The index is $2$ so automatically left cosets will coincide with right cosets.
If $\sigma\in A_n$ then $\sigma A_n=A_n=A_n\sigma$.
If $\sigma\notin A_n$ then $\sigma A_n=S_n\setminus A_n=A_n\sigma$.