Isomorphism is a concept that appears in various fields such as mathematics, computer science, and philosophy, but it generally refers to a type of mapping or relationship that preserves structure between two entities. In mathematics, particularly in abstract algebra, an isomorphism is a bijective (one-to-one and onto) function between two structures that preserves the operations defined on those structures.
Definition:
In the context of groups, a group isomorphism is a function \( f: G \to H \) between two groups \( (G, \cdot) \) and \( (H, *) \) such that:
- \( f \) is bijective.
- For all \( a, b \in G \), \( f(a \cdot b) = f(a) * f(b) \).
If such a function exists, we say that the two groups \( G \) and \( H \) are isomorphic, denoted \( G \cong H \).
Example:
Consider the following two groups:
- \( G = \mathbb{Z}/4\mathbb{Z} \) (the integers modulo 4 under addition), which contains the elements \( {0, 1, 2, 3} \).
- \( H = \mathbb{Z}_2 \times \mathbb{Z}_2 \) (the direct product of two copies of the integers modulo 2), which contains the elements \( {(0,0), (0,1), (1,0), (1,1)} \).
We can define a function \( f: G \to H \) as follows:
- \( f(0) = (0,0) \)
- \( f(1) = (0,1) \)
- \( f(2) = (1,0) \)
- \( f(3) = (1,1) \)
Verification of Isomorphism:
- Bijective: The function \( f \) is clearly a bijection, as each element in \( G \) maps to a unique element in \( H \) and vice versa.
- Operation Preservation: We need to check that \( f(a + b \mod 4) = f(a) + f(b) \mod 2 \times 2\).
Let's check some combinations:
-
For \( a = 1, b = 1 \):
- LHS: \( f(1 + 1 \mod 4) = f(2) = (1,0) \)
- RHS: \( f(1) + f(1) = (0,1) + (0,1) = (0+0, 1+1) = (0,0) \mod 2 = (0,0) \)
-
For \( a = 1, b = 2 \):
- LHS: \( f(1 + 2 \mod 4) = f(3) = (1,1) \)
- RHS: \( f(1) + f(2) = (0,1) + (1,0) = (0+1, 1+0) = (1,1) \)
You would continue checking similar combinations to verify that the operation preservation holds for all pairs.
Conclusion:
Since \( f \) is bijective and preserves the operation, we can conclude that the group \( \mathbb{Z}/4\mathbb{Z} \) is isomorphic to \( \mathbb{Z}_2 \times \mathbb{Z}_2 \), which means they have the same group structure, even though they consist of different elements.