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”.