Show directly that every \(k\)-cycle is a product of transpositions, and determine the minimum number of transpositions needed.
Solution
For a \(k\)-cycle \[ \sigma=(a_1\ a_2\ \cdots\ a_k), \] one has \[ \sigma =(a_1\ a_k)(a_1\ a_{k-1})\cdots(a_1\ a_2), \] where products are composed from right to left. Thus every \(k\)-cycle is a product of \(k-1\) transpositions.
This number is minimal. Multiplying a permutation by a transposition changes the number of cycles in its disjoint-cycle decomposition, counting fixed points, by exactly \(1\): it either joins two cycles or splits one cycle into two.
The identity on the support \(\{a_1,\ldots,a_k\}\) has \(k\) cycles, while a \(k\)-cycle has one cycle. Therefore at least \[ k-1 \] transpositions are required to pass from the identity to a \(k\)-cycle.
Hence the minimum number is exactly \[ \boxed{k-1}. \]