Quotient group

Let G be a group and let N be a normal subgroup of G. Then formula_1 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 formula_2.