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.