Permutation/Cycle/Definition

Cycle

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

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