Past exam of the groups and representations course of the University of Oxford / 2024 / 1 / b / Solution

Compose permutations from right to left. The three transpositions are ${\tau }_1=(23)$, ${\tau }_2=(13)$ and ${\tau }_3=(12)$. In cycle notation, direct composition givesThe complete symmetric group is thereforeConjugation relabels the entries of a cycle, so the conjugacy classes are preciselyTheir sizes are $1,3,2$, respectively. The differing orders of group elements also show that these three sets cannot merge under conjugation.