Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 3 / 5D / a

The order of a group element $g$ is the least positive integer $m$ for which $g^m=e$, if one exists. In a finite group, two terms of $e,g,g^2,\dots ,g^{|G|}$ must agree. If $g^i=g^j$ with $i<j$, cancellation gives $g^{j-i}=e$. Thus such positive integers exist and have a least member: every element of a finite group has finite order.

If $h=ag{a}^{-1}$, repeated cancellation shows $h^m=a{g}^m{a}^{-1}$ for every positive integer $m$. Consequently $h^m=e$ exactly when $g^m=e$. The least positive exponents are therefore equal:This proof also shows that conjugation preserves infinite order when the ambient group is not finite.