Prove that composition of isomorphisms is isomorphism

Question

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.

Answer 1

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 see that it's a bijection, note that since the individual maps are isomorphisms (and thus bijections), their composition is necessarily a bijection (there's no algebra here, just set theory).

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.