Homomorphism
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Definition
A homomorphism is a structure-preserving map between two algebraic structures of the same type.
For a map $\phi: A \to B$ to be a homomorphism, it must preserve the relevant operations.
In Different Contexts
| Structure | Homomorphism condition |
|---|---|
| Groups | $\phi(ab) = \phi(a)\phi(b)$ |
| Rings | $\phi(a + b) = \phi(a) + \phi(b)$ and $\phi(ab) = \phi(a)\phi(b)$ |
| Vector spaces | $\phi(ax + by) = a\phi(x) + b\phi(y)$ (linear map) |
| Modules | Preserves addition and scalar multiplication |
Key Properties
If $\phi: A \to B$ is a homomorphism:
- $\phi(e_A) = e_B$ (identity maps to identity)
- $\phi(a^{-1}) = \phi(a)^{-1}$ (inverses map to inverses)
- The image $\phi(A)$ is a substructure of $B$
- The kernel $\ker(\phi) = \{a : \phi(a) = e_B\}$ is a substructure of $A$
Special Cases
| Name | Definition |
|---|---|
| Monomorphism | Injective homomorphism |
| Epimorphism | Surjective homomorphism |
| isomorphism | Bijective homomorphism |
| Endomorphism | Homomorphism from a structure to itself |
| automorphism | Bijective endomorphism |
Intuition
A homomorphism “respects” the structure. It may lose information (not injective) or miss parts of the target (not surjective), but it never breaks the algebraic relationships.
Examples
- $\phi: \mathbb{Z} \to \mathbb{Z}_n$ given by $\phi(k) = k \mod n$ (group homomorphism)
- $\det: GL_n(\mathbb{R}) \to \mathbb{R}^*$ (the determinant is a group homomorphism)
- Any linear transformation is a vector space homomorphism
Related Concepts
- isomorphism — bijective homomorphism
- automorphism — isomorphism from a structure to itself
- Bijection — homomorphism + bijection = isomorphism