= Solution
For the <cycle-sum identity for Young–Jucys–Murphy elements>, let $C_m$ be the sum of all $m$-cycles on $\{1,\ldots,m\}$. Multiplying such a cycle by $(j\ m+1)$ inserts $m+1$ immediately after $j$ in that cycle, with our composition convention. Every $(m+1)$-cycle has a unique predecessor $j$ of $m+1$, so deleting $m+1$ recovers a unique term of $C_mX_{m+1}$. Since $X_2=(1\ 2)$, induction gives
$$
\boxed{X_2\cdots X_n=C_n.}
$$
For $n=1$ the empty product is the identity, also the unique one-cycle.
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