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?


  • $\begingroup$ "This means that the number $7$" is not in the group, since it doesn't have an inverse. ​ ​ $\endgroup$ – user991 Apr 15 '17 at 0:55
  • $\begingroup$ Ricky, I do not understand your comment. Do you agree with me that $Z_{21}^*$ is not a group? $\endgroup$ – Bob Apr 15 '17 at 1:11
  • $\begingroup$ No. ​ ​ ​ ​ ​ ​ ​ $\endgroup$ – user991 Apr 15 '17 at 12:48

$\mathbb Z_n^*$ 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.


for 21, imagine a torus of order 7 in one direction, and order 3 in the other. prime groups are cyclic.

  • $\begingroup$ Rob, from your answer I gather that you think $Z_{21}^{*}$ is an group. Assuming you are right (and I suspect you are) what is the inverse of $7$? $\endgroup$ – Bob Apr 15 '17 at 0:32

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