# Homomorphism

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In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "shape".

## Contents

### Definition

The definition of homomorphism depends on the type of algebraic structure under consideration. Particular definitions of homomorphism include the following:

The common theme is that a homomorphism is a function between two algebraic objects that respects the algebraic structure.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If G and H are groups, a homomorphism from G to H is a function ƒG → H such that $f(g_1 * g_2) = f(g_1) *' f(g_2)\,\!$ for any elements g1g2 ∈ G, where * denotes the operation in G and *' denotes the operation in H.