Understanding an implication involving symetric groups, alternating groups, and normal subgroups

Question

I am trying to understand the following statement:

Let's suppose that $\forall n, H \triangleleft A_n$ contains a $3$-cycle.

$A_n \triangleleft S_n$ has and index of $2$.

If $(abc) \in H \implies \exists (de) \in S_n/A_n$ s.t $(de)$ and $(abc)$ are disjoint and s.t. $(abc)(de) = (de)(abc)$

Here's what I don't understand

1. Why does there exist $(ed) \in S_n/A_n$?
2. Why are $(de)$ and $(abc)$ disjoint?
3. Why do they commute?

Answer 1

The mapping $S_n\rightarrow\{\pm 1\}:\pi\mapsto {\rm sgn}(\pi)$ is a group homomorphism, surjective if $n\geq 2$. The kernel is $A_n$ and so $A_n$ has index 2 in $S_n$.