What are the left and right cosets of $A_n$ in $S_n$?

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?

• Welcome to Mathematics Stack Exchange. $e$ and $(1,2)$ are representatives of cosets, and since $A_n$ is a normal subgroup, the right and left cosets are the same Commented Nov 5, 2019 at 1:49
• What do you mean by "visually see"? Commented Nov 5, 2019 at 16:30
• After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $\checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?. Commented Nov 5, 2019 at 16:41

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.