Quotient group

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

Simply, we can say that the quotient group of N in G as all elements in G that are not in N.

This set 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 $|G/H|=|G|/|H|$.