Cycle

Formal Definition

A permutation $\sigma \in S_n$ is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit.

Informal Definition

Replace this text with an informal definition.

Example(s)

The permutation $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 1 & 4 & 5 & 6 \end{pmatrix}$ of $S_6$ is a cycle because it has only one orbit with more than one element, namely $(1 2 3)$. The order of this cycle is $3$ because the largest orbit, $(1\ 2\ 3)$, contains $3$ elements.

Non-example(s)

The permutation $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 1 & 4 & 6 & 5 \end{pmatrix}$ of $S_6$ is NOT a cycle because it has two orbits with more than one element, namely $(1\ 2\ 3)$ and $(5\ 6)$.