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.

Also see

• Quotient Group Epimorphism is Epimorphism

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $24$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Quotient Groups
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 50.5$ Quotient groups
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Remark