Explicitly give the elements and structure of the group $S_n/A_n$, $n ≥ 5$.

Question

Explicitly give the elements and structure of the group $S_n/A_n$, $n ≥ 5$. I know that for $n≥5$, the alternating group $A_n$ is simple. Also for $n ≥ 5$, $S_n$ is an almost simple group, as it lies between the simple group $A_n$ and its group of automorphisms. I do not know how to "explicitly" write the structure of the group.

Answer 1

The elements of any quotient group $G/N$ are the cosets $gN$ (for different $g\in G$). How many cosets are there in your case? Write them all down, and you have explicitly given the elements.

Explicitly writing the structure means writing down a full multiplication table, or in some other, equivalent manner describe exactly how the group operation works. Since you have listed all the elements already, that shouldn't be too hard.

Answer 2

Hint:

For any two odd permutations $\sigma,\sigma'\in S_n$, we have $\sigma^{-1}\sigma'\in A_n$.