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.