Let G be a group and let N be a normal subgroup of G. Then ${\displaystyle G/N=\{gN:g\in {G}\))$ is the set of all cosets of N in G and is called the quotient group of N in G.

This group is used in the proof of Lagrange's Theorem, for instance. In fact, the proof of Lagrange's theorem establishes that if

G is finite, then ${\displaystyle |G/H|=|G|/|H|}$.