# Prove that composition of isomorphisms is isomorphism

I'm stuck!

Let $G, H$ and $K$ be three groups. Given $f: G \to H$ and $g: H\to K$ are isomorphisms, prove that the composition $g\circ f: G \to K$ is an isomorphism.

• Are you given that $f$ and $g$ are isomorphisms? Dec 10 '14 at 22:07
• Sorry yes they are Dec 10 '14 at 22:11
• You have two things then to show: $g\circ f$ is indeed a homomorphism (this follows from $f$ and $g$ being homomorphisms) and that it is bijective (this follows from $f$ and $g$ being bijections). I'll leave the details to you. Dec 10 '14 at 22:13
• @Cameron: oog, no. Show that if $f$ and $g$ have inverses then $gf$ has an inverse. Dec 11 '14 at 1:58

First, what needs to be done in order to show that the mapping $g\circ f$ is an isomorphism? Well, you can simply make sure it first obeys the multiplicativeness law $(g\circ f)(ab)=(g\circ f)(a)(g\circ f)(b)$. It should also be a bijection.
To show the other condition, start by letting $x,y\in G$ and considering $(g\circ f)(xy)$. From there you are able somehow to end up with the RHS of the homomorphism law.