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For a permutation $\sigma$ of a finite set, the cycle containing $x$ is its orbit under the iterates of $\sigma$. Its length $k$ is the least positive integer with ${\sigma }^{k}x=x$; finiteness and invertibility guarantee such a return. Two cycles that meet are the same, because any iterate of a common point belongs to both. Thus the cycles partition the set. In cycle notation, the displayed cycle sends each listed element to the next and the last one back to $x$.

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.