A cycle $[a_1,a_2,\ldots,a_n]$ is an arrangement of $n$ objects such that it "wraps around" because it is joined to its beginning, more formally $$\begin{align}[a_1,a_2,a_3\ldots,a_{n-1},a_{n}] &=[a_2,a_3\ldots,a_{n-1},a_{n},a_1]\nonumber\\ &=[a_3,a_4\ldots,a_{n-1},a_{n},a_1,a_2]\nonumber\\ &\vdots\nonumber\\ &=[a_{n},a_1,a_2\ldots,a_{n-2},a_{n-1}]\nonumber\end{align}$$
Definitions: 1
Explanations: 2
Proofs: 3 4