# 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”.