What are the left and right cosets of $A_n$ in $S_n$?
I know that $A_n$ is the set of all even permutations and $S_n$ is the set of all permutations.
I was looking online and found that since $A_n$ is normal, the other coset is $S_n - A_n$(since cosets partition a group).
For $A_4$ in $S_4$, it's pretty visual because you can clearly see $e$ and $(1,2)$ are the left cosets and $e$ is the right coset.
Is there any way to visually see it for $A_n$ in $S_n$ by listing?
Thanks in advance.
Since $A_n$ is normal in $S_n$, we have for all $\sigma\in S_n$ that $\sigma A_n\sigma^{-1}=A_n$; hence $\sigma A_n=A_n\sigma$; that is, the left and right cosets are the same.
Also, here $e$ and $(1,2)$ are not cosets; they are representatives of cosets. The cosets look like this:
$$\sigma A_n=\{\sigma\tau\mid \tau\in A_n\}$$
for each $\sigma\in S_n$ up to the representative.
To see $A_n$ in $S_n$ visually, try reading "Visual Group Theory," by Carter.