I'm a bit confused as to what I'm supposed to be doing in the following introductory abstract algebra question.
I am asked to write all of the elements of $\mathbb{Z}^{x}_5$,$\mathbb{Z}^{x}_6$, $\mathbb{Z}^{x}_8$, and $\mathbb{Z}^{x}_{20}$
Is this as simple as writing out the elements of $\mathbb{Z}_5$ under modular multiplication? I don't think that is the case as the result isn't a group.
So my other thought is that I would write out the 5th roots of unity?
That $\mathbb{Z}^{x}_5$={1,$e^{\frac{2\pi *i}{5}}$, $e^{\frac{4\pi *i}{5}}$, $e^{\frac{6\pi *i}{5}}$, $e^{\frac{8\pi *i}{5}}$}={1,$e^{\frac{2\pi *i}{5}}$, $e^{\frac{4\pi *i}{5}}$, $e^{-\frac{4\pi *i}{5}}$, $e^{-\frac{2\pi *i}{5}}$}
Am I on the right track with this?
The set $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is the set of equivalence classes $\bar{a}$ which have multiplicative inverses mod $n$. Moreover $$\text{$(\mathbb{Z}/n\mathbb{Z})^{\times}=\{\bar{a}\in \mathbb{Z}/n\mathbb{Z}\ \mid$ gcd $(a,n)=1$}\}.$$