A Question about Notations and Groups
$\mathbb Z_n^*$ 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.