Subgroups of order 5 in $S_5$ are transitive

Show that every subgroup of order $5$ in $S_5$ acts transitively on $\{1,2,3,4,5\}$.

Solution

Let $H\le S_5$ have order $5$. Then $H$ is cyclic, say \[ H=\langle\sigma\rangle \] with $\sigma\ne e$. Since $\sigma$ has order $5$, its cycle decomposition in $S_5$ must consist of a single $5$-cycle. Therefore the orbit of any point under $\langle\sigma\rangle$ has size $5$ and is the whole set.