Left cosets of $A_6$ in $S_6$ Which may be the all of left cosets of $A_6$ in $S_6$?
$\{A_6,(156)A_6\},\{A_6,(34)A_6\},\{A_6,(42)(35)A_6\}, \{A_6,(46523)A_6\}$ or $\{A_6\}$
I dont understand why the answer is $\{A_6,(34)A_6\}$ .I know if we multiply $A_6$ elements in $S_6$ with $A_6$ we get $A_6$ itself. why  multiplying $S_6-A_6$ element with $A_6$ is equal to $(36)A_6$?

$A_{6}$ is the group of all even permutations of $6$ elements. If you multiply that set by an even permutation, you get $A_{6}$ back, but if you multiply by any odd permutation you get every odd permutation (you need to prove that every distinct even permutation is sent to a distinct odd permutation). So the problem is just: which of those 5 permutations is odd? I leave that to you.
