# Homomorphism

## Structure-preserving map between two algebraic structures of the same type / From Wikipedia, the free encyclopedia

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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). The word *homomorphism* comes from the Ancient Greek language: ὁμός (*homos*) meaning "same" and μορφή (*morphe*) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to ὁμός meaning "same".^{[1]} The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).^{[2]}

Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra.

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. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.