> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/400-abstract-algebra/terms/homomorphism.md).

# Homomorphism

<https://en.wikipedia.org/wiki/Homomorphism>

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

In every category of algebraic structures, an isomorphism is a homomorphism that is a bijection.

<https://en.wikipedia.org/wiki/Homomorphism>

**Homomorphism** is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, two vector spaces).

The concept of homomorphism has been generalized, under the name of *morphism*, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of *category theory*.

A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. Each of those can be defined in a way that may be generalized to any class of morphisms.