Isomorphism

Isomorphism is a very general concept which appears in several areas of mathematics. Formally, an isomorphism is Bijective Morphism. Informally, an isomorphism is a map which preserves sets and relations among elements.

A space isomorphism is a Vector Space in which addition and scalar multiplication are preserved. An isomorphism of a Topological Space is called a Homeomorphism.

Two groups $G_1$ and $G_2$ with binary operators $+$ and $\times$ are isomorphic if there exists a map $f:G_1\mapsto G_2$ which satisfies

$$f(x+y)=f(x)\times f(y).$$

An isomorphism preserves the identities and inverses of a Group. A Group which is isomorphic to itself is called an Automorphism.

See also Automorphism, Ax-Kochen Isomorphism Theorem, Homeomorphism, Morphism