Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 5 / a / iii

The order of a group element $g$ is the smallest positive integer $d$ with $g^d=e$. Such an integer exists in a finite group: two powers in $e,g,g^2,\dots$ must agree, and cancellation gives a positive power equal to $e$.

By minimality, $e,g,\dots ,g^{d-1}$ are distinct. Every integer power of $g$ is one of them, by division of its exponent by $d$. They form the cyclic subgroupApplying Lagrange's theorem to this subgroup gives