Isomorphism

isomorphism, in modern algebra, a one-to-one correspondence (mapping) between two sets that preserves binary relationships between elements of the sets. For example, the set of natural numbers can be mapped onto the set of even natural numbers by multiplying each natural number by 2. The binary operation of adding two numbers is preserved—that is, adding two natural numbers and then multiplying the sum by 2 gives the same result as multiplying each natural number by 2 and then adding the products together—so the sets are isomorphic for addition.

In symbols, let A and B be sets with elements a_n and b_m, respectively. Furthermore, let ⊕ and ⊗ indicate their respective binary operations, which operate on any two elements from a set and may be different. If there exists a mapping f such that f(a_j ⊕ a_k) = f(a_j) ⊗ f(a_k) and its inverse mapping f⁻¹ such that f⁻¹(b_r ⊗ b_s) = f⁻¹(b_r) ⊕ f⁻¹(b_s), then the sets are isomorphic and f and its inverse are isomorphisms. If the sets A and B are the same, f is called an automorphism.

Because an isomorphism preserves some structural aspect of a set or mathematical group, it is often used to map a complicated set onto a simpler or better-known set in order to establish the original set’s properties. Isomorphisms are one of the subjects studied in group theory.