Exercise 8

Let $A$, $B$, and $C$ be non-empty sets. Let $f:A\to B$ and $g:B\to C$ be bijections. Let $h=g\circ f$, where $\circ$ represents function composition. That is, $h:A\to C,x\mapsto g(f(x))$.

1. Show that $h$ is a bijection.

2. If $A$, $B$, and $C$ are groups, and $f$ and $g$ are isomorphisms, show that $h$ is an isomorphism.