Definition 6

The quotient of a group with operators G by the equivalence relation defined by a normal subgroup H of G, with the quotient structure, is called the quotient group with operators of G by H and is denoted by $G/H$. The canonical mapping $G \to G/H$ is called a canonical homomorphism

Let G be a group and H a normal subgroup of G. The quotient $G/H$, with its group structure, is called the quotient group of G by H. For a mapping from $G/H$ to a group with operators to be a homomorphism of groups with operators, it is necessary and sufficient that its composition with the canonical mapping of G onto $G/H$ be one: this justifies the name "quotient group" (Set Theory, IV, $\S$ 2, no. 6).

The equivalence relation defined by a normal stable subgroup of G is denoted by $x \equiv y \pmod{H}$ or $x \equiv y(H)$.