Definition:Quotient Epimorphism/Group

Definition
Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

The mapping $q: G \to G / N$ defined as:
 * $\forall x \in G: q \left({x}\right) = x N$

is known as the quotient (group) epimorphism from $G$ to $G / N$.

Also known as
The quotient (group) epimorphism is also known as:
 * the quotient (group) morphism
 * the natural (group) epimorphism
 * the natural (group) morphism
 * the natural (group) homomorphism
 * the canonical (group) epimorphism
 * the canonical (group) morphism.

In all of the above, the specifier group is usually not used unless it is necessary to distinguish it from a quotient ring epimorphism.

Also see

 * Quotient Group Epimorphism is Epimorphism