Permutation/Cycle/Definition

Let ${\displaystyle {}M}$ be a finite set and let ${\displaystyle {}\pi }$ be a permutation on ${\displaystyle {}M}$. Then ${\displaystyle {}\pi }$ is called a cycle of order ${\displaystyle {}r}$ (or of length ${\displaystyle {}r}$), if there exists a subset ${\displaystyle {}Z\subseteq M}$, containing ${\displaystyle {}r}$ elements and such that ${\displaystyle {}\pi }$ is on ${\displaystyle {}M\setminus Z}$ the identity and such that ${\displaystyle {}\pi }$ commutes the elements of ${\displaystyle {}Z}$ in a cyclic way. If ${\displaystyle {}Z=\{z,\pi (z),\,\pi ^{2}(z),\ldots ,\pi ^{r-1}(z)\}}$, then we write

${\displaystyle {}\pi =\langle z,\pi (z),\,\pi ^{2}(z),\ldots ,\pi ^{r-1}(z)\rangle \,.}$