Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 5D (Groups)

The order of $g$ is the least positive $r$ with $g^r=e$. Division $n=qr+s$ shows $g^n=e$ forces $s=0$. Lagrange theorem follows because left cosets of a subgroup partition $G$ into equal blocks; applying it to $⟨g⟩$ shows $|g|\mid |G|$. Coset multiplication makes $G/N$ a group when $N◃G$. If $x\in N$, then $x^n=e$. Conversely, if $x^n=e$, the order of $xN$ divides both $m$ and $n$, hence is one, so $x\in N$.

Every $b^m$ lies in $N$ because $(bN{)}^m=N$. Conversely choose $r$ with $mr\equiv 1(\mathrm{mod}n)$; for $a\in N$, $a=(a^r{)}^m$ because $a^n=e$.