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 $(36)A_6$?

There are only two cosets of $A_6$ in $S_6$, since the alternating group $A_n$ has index $2$ in $S_n$ for any $n>1$. 
It has index $2$ because  $A_n$ is the kernel of the signature morphism: $\;\varepsilon\colon S_n\to \{+1,-1\}$. Hence $A_n$ is normal in $S_n$, and these cosets are left as well as right cosets.
The cosets are, namely:


*

*$A_n$ itself; its  elements have signature $+1$;

*the coset of any odd permutation, like the transpostion $(34)$; its elements map to $-1$.

