A Question about Notations and Groups Please consider the following question:
Determine the order of all the elements of the following multiplicative groups. You can write a C or Java program to do this.
a. $Z_{21}^*$
b. $Z_{23}^*$  

Now note that $21$ is not a prime number and since the question refers to a
multiplicative group, I would expect that means the operator for the group is
multiplication mod 21. This means that the number $7$ should have an inverse. That is, there should be a number x, such that $(7x)\,mod\,\,21\,\,=\,1$. I claim that no such $x$ exists. What am I missing?

Bob
$\mathbb Z_n^*$ is the subset of $\mathbb Z_n$ with elements relatively prime to $n$ (or equivalently: with an inverse under multiplication modulo $n$). That's a group under multiplication modulo $n$. It has $\varphi(n)$ elements, where $\varphi$ is the Euler totient.

It follows that $7\not\in\mathbb Z_{21}^*$; that's what's missed in the question.