The composition of two group homomorphisms is a homomorphism

by Patrick Stevens Jun 14 2016 updated Jun 14 2016

The collection of group homomorphisms is closed under composition.

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.