Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 9 / i

For a subgroup $H$ of a finite group $G$, Lagrange's theorem statesHere $[G:H]$ is the number of distinct left cosets. Every coset has $|H|$ members, and the cosets partition $G$, proving the identity. In particular the order of a subgroup divides the order of the group. Divisibility is generally necessary, not sufficient, for a subgroup of a given order to exist.