Definition

Kernel and Image

Lie Groups and Lie Algebras · lie-groups.tex

For a homomorphism $\phi : G \to H$ (Definition~Group Homomorphism), the kernel and image are \[ \ker \phi := \{ g \in G \mid \phi(g) = e_H \}, \qquad \operatorname{im} \phi := \{ \phi(g) \mid g \in G \}. \]

Depends on

Group Homomorphism

Used in

Proposition
First Isomorphism Theorem
Universal Cover of a Lie Group

Dependency Graph

flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 ne087180c["Group Homomorphism"] n6fd9cebe["Kernel and Image"]:::current n8aabc776["Proposition"] n856a1ed3["First Isomorphism Theorem"] n738e8542["Universal Cover of a Lie Group"] ne087180c --> n6fd9cebe n6fd9cebe --> n8aabc776 n6fd9cebe --> n856a1ed3 n6fd9cebe --> n738e8542 click ne087180c "../objects/e087180c.html" "_self" click n8aabc776 "../objects/8aabc776.html" "_self" click n856a1ed3 "../objects/856a1ed3.html" "_self" click n738e8542 "../objects/738e8542.html" "_self"