Definition:Kernel of Group Homomorphism

Definition

Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:

$\ker \left({\phi}\right) = \left\{{x \in G: \phi \left({x}\right) = e_H}\right\}$

where $e_H$ is the identity of $H$.

That is, $\ker \left({\phi}\right)$ is the subset of $G$ that maps to the identity of $H$.

Also see

• Identity in Kernel of Group Homomorphism where it is shown that $e_G \in \ker \left({\phi}\right)$ where $e_G$ is the identity of $G$.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 7.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 12$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.10$: Theorem $22$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 65$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 47$
• John F. Humphreys: A Course in Group Theory (1996): $\S 8$: Definition $8.12$