2.15 Powers and orders ‣ Chapter 2 Sets and functions ‣ MATH0005 Algebra 1‣ Chapter 2 Sets and functions ‣ MATH0005 Algebra 1 (2024)

2.15.1 Powers of a permutation

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

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

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

2.15.2 Order of a permutation

For example, let

You should check that $s^2\ne \mathrm{id}$ but $s^3=\mathrm{id}$, so the order of $s$ is 3, and that $t\ne \mathrm{id}$ but $t^2=\mathrm{id}$ so the order of $t$ is2.

2.15.3 Order of an $m$-cycle