This is an exercise 3, chapter 8 from Stein's Fourier Analysis. In this chapter, the author deals with finite abelian groups and Fourier analysis on them. The problem is the following: Let $p$ and $q$ be relatively prime. Prove that $\mathbb Z^*(p) \times\mathbb Z^*(q)$ is isomorphic to $\mathbb Z^*(pq)$. (Here $\mathbb Z^*$ means the multiplicative group.)
I know that for additive groups, $\mathbb Z(a) \times\mathbb Z(b)$ is isomorphic to $\mathbb Z(ab)$ if $a$ and $b$ are relatively prime. Indeed, we can make an isomorphism, $$\phi:(1,1)\mapsto 1$$ But for multiplicative groups I don't know how to make an isomorphism between them. For example, $\mathbb Z^*(5)=\{1,2,3,4\}$ and $\mathbb Z^*(7)=\{1,2,3,4,5,6\}$. Here how can we define an isomorphism mapping some element from $\mathbb Z^*(7)$ $\times\mathbb Z^*(5)$ to $33\in\mathbb Z^*(35)$?
