Congruence Relation - Relation With Homomorphisms

Relation With Homomorphisms

If ƒ: A → B is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear map between vector spaces), then the relation ≡ defined by

a1 ≡ a2 if and only if ƒ(a1) = ƒ(a2)

is a congruence relation. By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

Read more about this topic:  Congruence Relation

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