Let $S(X)$ denote the group of permutations of a finite set $X$. Show that every permutation $\sigma \in S(X)$ can be written as a product of disjoint cycles. Explain briefly why two permutations in $S(X)$ are conjugate if and only if, when they are written as the product of disjoint cycles, they have the same number of cycles of length $n$ for each possible value of $n$.

Let $\ell(\sigma)$ denote the number of disjoint cycles, including 1-cycles, required when $\sigma$ is written as a product of disjoint cycles. Let $\tau$ be a transposition in $S(X)$ and $\sigma$ any permutation in $S(X)$. Prove that $\ell(\tau \sigma)=\ell(\sigma) \pm 1$.