A question on the intuition of decomposition of the element of symmetry group

Question

Any element of symmetry group $S_{n}$ can be decomposed as products of transpositions. Any m-cycle can be decomposed as m-1 transposition products. How should I think of this decomposition? Is there any intuitive way of thinking this decomposition without memorizing the formula of decomposition or is there an intuitive way to derive the decomposition formula? I knew from the book the formula for decomposition was given explicitly.

Answer 1

Take $m$ pieces of paper labelled $1$, $2$, $3$, … $m$. Place them on the table in the following order: $$m,1,2,3,4,5,…,m-2,m-1 $$ That represents an $m$-cycle. Now, how do you put them back in order? First transpose $m$ and $1$: $$1,m,2,3,4,5,…,m-2,m-1 $$ Next transpose $m$ and $2$: $$1,2,m,3,4,5,…,m-2,m-1 $$ Next transpose $m$ and $3$: $$1,2,3,m,4,5,…,m-2,m-1 $$ And so on… after $m-2$ of these transpositions you get to $$1,2,3,4,5,…,m-2,m,m-1 $$ And one last transposition, the $m-1$st one: transpose $m$ and $m-1$: $$1,2,3,4,5,…,m-2,m-1,m $$