Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 8 / iii

The order of a group element $g$ is the least positive integer $q$ such that $g^q=e$, or infinity if there is no such integer. In a finite group such a $q$ exists, because two powers among $e,g,g^2,\dots$ must coincide.

The powers $e,g,\dots ,g^{q-1}$ are distinct: a repetition with exponents differing by less than $q$ would contradict its minimality. Division of any exponent by $q$ shows that these are all the elements of the cyclic subgroup $⟨g⟩$. Thus $|⟨g⟩|=q$. Applying Lagrange's theorem givesIn particular the identity has order one, and every nonidentity element has order greater than one. The divisibility assertion here concerns finite group order.