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In abstract algebra, a homomorphism is a structurepreserving 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 and illustration
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 ƒ: G → H such that for any elements g_{1}, g_{2} ∈ G, where * denotes the operation in G and *' denotes the operation in H.
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