Definition
Group Homomorphism
Let $(G, \cdot)$ and $(H, \ast)$ be groups in the sense of Definition~Group (Classical).
A map $\phi : G \to H$ is a group homomorphism if
\[
\phi(a \cdot b) = \phi(a) \ast \phi(b) \quad for all a, b \in G.
\]
If $\phi$ is also a bijection it is called an isomorphism, and we write $G \cong H$.
Depends on
Used in
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
n71746aac["Group (Classical)"]
ne087180c["Group Homomorphism"]:::current
n6fd9cebe["Kernel and Image"]
n8aabc776["Proposition"]
n856a1ed3["First Isomorphism Theorem"]
n738e8542["Universal Cover of a Lie Group"]
nca6bd350["Representation"]
nf70e3a35["Morphism of Representations"]
n71746aac --> ne087180c
ne087180c --> n6fd9cebe
ne087180c --> n8aabc776
ne087180c --> n856a1ed3
ne087180c --> n738e8542
ne087180c --> nca6bd350
ne087180c --> nf70e3a35
click n71746aac "../objects/71746aac.html" "_self"
click n6fd9cebe "../objects/6fd9cebe.html" "_self"
click n8aabc776 "../objects/8aabc776.html" "_self"
click n856a1ed3 "../objects/856a1ed3.html" "_self"
click n738e8542 "../objects/738e8542.html" "_self"
click nca6bd350 "../objects/ca6bd350.html" "_self"
click nf70e3a35 "../objects/f70e3a35.html" "_self"