Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 6 / a

Lagrange's theorem states that for a subgroup $K$ of a finite group $G$,in particular, $|K|$ divides $|G|$. Here $[G:K]$ is the subgroup index, the number of left cosets of $K$.

The order of a group element $g\in G$ is the least positive integer $k$ such that $g^k=e$. It exists because $G$ is finite. The cyclic subgroup $⟨g⟩$ consists of exactly $e,g,\dots ,g^{k-1}$, so has order $k$. Applying Lagrange's theorem to this subgroup gives