Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 7D / a / Solution

Lagrange's theorem says that, for a finite group $G$ and subgroup $H$,Here $[G:H]$ is the number of left cosets. Every $g\in G$ lies in $gH$. If $gH$ and $g^{\prime }H$ meet, write $gh=g^{\prime }{h}^{\prime }$; then $g^{\prime }=gh(h^{\prime }{)}^{-1}$, so $g^{\prime }H=gH$. Hence the left cosets partition $G$. The bijection $h\mapsto gh$ makes each coset have $|H|$ elements. Adding their cardinalities proves Lagrange's theorem.