About motivation of Lang's Proof $S_n$ is not solvable for $n\geq 5$.

Question

In Lang, he proves that $S_n$ is not solvable if $n\geq 5$ by using following observation.

If $N\unlhd H\leq G$, $H$ contains every $3$-cycle, and if $H/N$ is abelian, then $H$ contains every $3$-cycle.

Where is this observation coming from? Why $3$-cycle, not $2$-cycle or some other $m$-cycle?