Every Disjoint Cycle Can Be Written As A Product Of Transpositions

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Problem 65B (optional) (Every Disjoint Cycle Can Be Written As A Product Of Transpositions)

Start by convincing yourself that $(1,2,3,4,5)=(1,5)(1,4)(1,3)(1,2)$. This shows how to write a 5-cycle as a product of transpositions (2-cycles).

1. Find another way to write $(1,2,3,4,5)$ as a product of transpositions. This shows that there are multiple ways to write a cycle as a product of transpositions.
2. Suppose $m,n\in \mathbb{N}$ with $m\geq n$. Also suppose that $\alpha = (a_1,a_2,a_3, \ldots, a_n)\in S_m$ is a disjoint cycle. Give a way to rewrite $\alpha$ as a product of 2-cycles.

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