Past exam of the mathematics course of the University of Oxford / 2014 / Prelims / Mathematics I / 6 / b / ii

By the preceding relabelling formula, every conjugate of $(123)$ is a three-cycle, and any three-cycle on four symbols can be obtained by a suitable relabelling. ThereforeThere are eight elements, agreeing with the count $\left(\genfrac{}{}{0ex}{}{4}{3}\right)\cdot 2$.

Likewise, conjugation preserves the pattern of two disjoint transpositions, and every partition of four symbols into two pairs is realized by relabelling. ThusThese have the respective cycle types $(3,1)$ and $(2,2)$ in the symmetric group $S_4$.