Question
Define isomorphism with an example
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GPT-4o mini
Isomorphism is a mathematical term used to describe a relationship between two structures that show a one-to-one correspondence between their elements and preserve the operations defined on those structures. In simpler terms, two structures are isomorphic if they can be transformed into each other in a way that maintains their properties.
Isomorphisms can occur in various branches of mathematics, including algebra, graph theory, and topology. The key feature of an isomorphism is that it creates a "mapping" or "correspondence" that is reversible and structure-preserving.
### Example
**Example in Group Theory:**
Consider the groups \( G = \mathbb{Z}/4\mathbb{Z} \) (the integers modulo 4 under addition) and \( H = \{ e, a, a^2, a^3 \} \), where \( a^n \) represents the powers of some element \( a \) in group \( H \) (let's say it is a cyclic group of order 4).
The operation in both groups is addition modulo 4 for \( G \) and cyclic group multiplication for \( H \).
We can define a mapping \( \phi: G \to H \) as follows:
- \( \phi(0) = e \)
- \( \phi(1) = a \)
- \( \phi(2) = a^2 \)
- \( \phi(3) = a^3 \)
This mapping is a bijection (one-to-one and onto), and it preserves group operation:
- \( \phi(x + y \mod 4) = \phi(x) \cdot \phi(y) \)
For example:
- \( \phi(1 + 2 \mod 4) = \phi(3) = a^3 \)
- \( \phi(1) \cdot \phi(2) = a \cdot a^2 = a^3 \)
Since the mapping preserves both the structure and operations of the groups, \( G \) and \( H \) are isomorphic, often written as \( G \cong H \).
This shows that \( G \) and \( H \) are essentially the same group, even though they are represented differently, indicating the concept of isomorphism in mathematics.
Isomorphisms can occur in various branches of mathematics, including algebra, graph theory, and topology. The key feature of an isomorphism is that it creates a "mapping" or "correspondence" that is reversible and structure-preserving.
### Example
**Example in Group Theory:**
Consider the groups \( G = \mathbb{Z}/4\mathbb{Z} \) (the integers modulo 4 under addition) and \( H = \{ e, a, a^2, a^3 \} \), where \( a^n \) represents the powers of some element \( a \) in group \( H \) (let's say it is a cyclic group of order 4).
The operation in both groups is addition modulo 4 for \( G \) and cyclic group multiplication for \( H \).
We can define a mapping \( \phi: G \to H \) as follows:
- \( \phi(0) = e \)
- \( \phi(1) = a \)
- \( \phi(2) = a^2 \)
- \( \phi(3) = a^3 \)
This mapping is a bijection (one-to-one and onto), and it preserves group operation:
- \( \phi(x + y \mod 4) = \phi(x) \cdot \phi(y) \)
For example:
- \( \phi(1 + 2 \mod 4) = \phi(3) = a^3 \)
- \( \phi(1) \cdot \phi(2) = a \cdot a^2 = a^3 \)
Since the mapping preserves both the structure and operations of the groups, \( G \) and \( H \) are isomorphic, often written as \( G \cong H \).
This shows that \( G \) and \( H \) are essentially the same group, even though they are represented differently, indicating the concept of isomorphism in mathematics.