# 2.14 Powers and orders

## 2.14.1 Powers of a permutation

Since the composition of two permutations is another permutation, we can form powers of a permutation by composing it with itself some number of times.

###### Definition 2.14.1.

Let $s$ be a permutation and let $m$ be an integer. Then

 $s^{m}=\begin{cases}s\cdots s\;(m\text{ times})&m>0\\ \operatorname{id}&m=0\\ s^{-1}\cdots s^{-1}\;(-m\text{ times})&m<0\end{cases}$

It’s tedious but straightforward to check that for any integers $a$, $b$,

• $s^{a}\circ s^{b}=s^{a+b}$, and

• $(s^{a})^{b}=s^{ab}$

so that some of the usual exponent laws for real numbers hold for composing permutations. The two facts above are called the exponent laws for permutations.

## 2.14.2 Order of a permutation

###### Definition 2.14.2.

The order of a permutation $\sigma$, written $o(\sigma)$, is the smallest strictly positive number $n$ such that $\sigma^{n}=\operatorname{id}$.

For example, let

 $\displaystyle s$ $\displaystyle=\begin{pmatrix}1&2&3\\ 2&3&1\end{pmatrix}$ $\displaystyle t$ $\displaystyle=\begin{pmatrix}1&2&3\\ 2&1&3\end{pmatrix}$

You should check that $s^{2}\neq\operatorname{id}$ but $s^{3}=\operatorname{id}$, so the order of $s$ is 3, and that $t\neq\operatorname{id}$ but $t^{2}=\operatorname{id}$ so the order of $t$ is 2.

## 2.14.3 Order of an $m$-cycle

###### Lemma 2.14.1.

The order of an $m$-cycle is $m$.

###### Proof.

Let the $m$-cycle be $a=(a_{1},\ldots,a_{m})$. If $r then $a^{r}(a_{1})=a_{r+1}\neq a_{1}$, so $a^{r}\neq\operatorname{id}$. On the other hand $a^{m}(a_{1})=a(a_{m})=a_{1}$ and in general $a^{m}(a_{i})=a^{i}(a^{m-i}(a_{i}))=a^{i}(a_{1})=a_{i}$, so $a^{m}=\operatorname{id}$. ∎