Cycle notation

Remarks

The same cycle has $k$ different notations, because you can start the list at any entry: $(1\ 2\ 3) = (2\ 3\ 1) = (3\ 1\ 2)$. The inverse of $(a_1\ a_2\ \dots\ a_k)$ is $(a_k\ \dots\ a_2\ a_1)$. Each permutation of a finite set is a product of disjoint cycles, so cycle notation can show each permutation. For example, in $S_5$ the permutation with $1 \mapsto 3$, $3 \mapsto 1$, $2 \mapsto 4$, $4 \mapsto 5$, $5 \mapsto 2$ is $(1\ 3)(2\ 4\ 5)$.