C3 - cycle

(a_{1}, a_{2}, \dots, a_{m}) (k) = {\begin{cases} a_{i + 1} & if k = a_{i}, 1 \leq a_{i} \leq m, \\ a_{1} & if k = a_{m}, \\ k & if k \notin {a_{1}, a_{2,} \dots, m} . \end{cases}

[\begin{matrix} a_{1} & a_{2} & \dots & a_{i} & \dots & a_{m - 1} & a_{m} \\ a_{2} & a_{3} & \dots & a_{i + 1} & \dots & a_{m} & a_{1} \end{matrix}]

α = [\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 4 & 6 & 5 & 3 \end{matrix}]

α = (12) (346) (5)

cycles can be multiplied, $α β$ , and the results can be expressed as disjoint cycles
as function composition is done for right to left, so is cycle multiplication
any symbol not contained in a cycle is fixed, ie. $(234)$ fixes (1)
conventionally, single element cycles are omitted as they are mapping to themselves, ie. $(1) (234) (56) \equiv (234) (56)$
identity permutations can be written as just one cycle, eg. $ϵ = (5)$ or $(1)$
note: some conventions may also use left to right composition