Isomorphism
★★★★☆Undergraduate
📖Definition
An isomorphism is a bijective homomorphism. If two structures are isomorphic, they are algebraically identical. It represents essential structural sameness.
📐Formulas
φ: G → H isomorphism ⇔ φ bijective homomorphism
Definition of isomorphism
G ≅ H
G and H are isomorphic
G / ker(φ) ≅ im(φ)
First Isomorphism Theorem
✏️Examples
예제 1
Show (ℤ, +) and (2ℤ, +) are isomorphic.
예제 2
Show all groups of order 2 are isomorphic.
📜History
Discovered by: Emmy Noether (1920s)
Noether generalized isomorphism theorems and laid foundations of modern algebra.
⚡Applications
Algebra
Structure classification
Graph Theory
Graph isomorphism
Computer Science
Data structure equivalence
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