Exercise 6.4.12

Proof. Let $\sigma \in S_p$ a permutation of order p. Write $\sigma ={\sigma }_1\cdots {\sigma }_r({\sigma }_i\ne e)$ the cycle decomposition of $\sigma$. Let $d_i=|{\sigma }_i|$ the order of ${\sigma }_i$ in $S_n$. The order of $\sigma$ is the lcm of the orders $d_i$ (see Ex. 6).

As $d_i\mid p,i=1,\dots ,r$, and $d_i\ne 1$, where $p$ is prime, $d_i=p$. The cycles ${\sigma }_i$ being disjoint, as $d_i=|{\sigma }_i|=\mathrm{length}({\sigma }_i)$, $d_1+\cdots +d_r\le p$, thus $r{d}_1=\mathit{pr}\le p$, so $r=1$.

Conclusion : $\sigma ={\sigma }_1$ is a $p$-cycle. □