Isomorphism
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Definition
An isomorphism is a bijection between two mathematical structures that preserves the structure’s operations.
Two structures are isomorphic if an isomorphism exists between them — they are “the same” up to relabeling.
In Different Contexts
| Context | Isomorphism preserves |
|---|---|
| Sets | Just a bijection (no extra structure) |
| Groups | $\phi(ab) = \phi(a)\phi(b)$ |
| Rings | Addition and multiplication |
| Vector spaces | Linear combinations |
| Topological spaces | Open sets (homeomorphism) |
| Metric spaces | Distances (isometry) |
| Inner product spaces | Inner products (unitary isomorphism) |
Key Idea
If $\phi: A \to B$ is an isomorphism, then:
- Bijection: $\phi$ is one-to-one and onto
- Structure-preserving: Operations in $A$ correspond to operations in $B$
- Invertible: $\phi^{-1}: B \to A$ is also an isomorphism
Examples
- $(\mathbb{R}, +) \cong (\mathbb{R}^+, \times)$ via $\phi(x) = e^x$
- The Fourier transform is a unitary isomorphism $L^2(\mathbb{R}) \to L^2(\mathbb{R})$
- Any two finite-dimensional vector spaces of the same dimension are isomorphic
Intuition
Isomorphic structures are “mathematically identical” — they have the same abstract properties. The specific elements don’t matter; only their relationships do.
Related Concepts
- Bijection — an isomorphism is a structure-preserving bijection
- homomorphism — structure-preserving map (not necessarily bijective)
- automorphism — isomorphism from a structure to itself