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 $g\left(f\right(x\left)\right)g\left(f\right(y\left)\right)=g\left(f\right(x\left)f\right(y\left)\right)$ since $g$ is a homomorphism; that is $g\left(f\right(xy\left)\right)$ because $f$ is a homomorphism.