# The composition of two group homomorphisms is a homomorphism

Given two group homomorphisms $$f: G \to H$$ and $$g: H \to K$$, the composition $$gf: G \to K$$ is also a homomorphism.

To prove this, note that $$g(f(x)) g(f(y)) = g(f(x) f(y))$$ since $$g$$ is a homomorphism; that is $$g(f(xy))$$ because $$f$$ is a homomorphism.

Parents:

• Group homomorphism

A group homomorphism is a “function between groups” that “respects the group structure”.