# 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 $q_N: G \to G / N$ defined as:

$\forall x \in G: \map {q_N} x = x N$

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
• the projection
• the quotient map.

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