algebra Definition Group Homomorphism A group homomorphism φ:(G,∘)→(H,∗) is a morphism in algebraic structures (such as groups, rings or vector spaces) that maps from a group (G,∘) to a group (H,∗) such that: ∀x,y∈G:φ(x∘y)=φ(x)∗φ(y)