I'm trying to solve a problem which asks to find how many elements the multiplicative group of a ring $R$ has. It is defined as follows: $R= \Big\{ \left(\begin{matrix} a & b \\ -b & a \\ \end{matrix}\right) | a,b \in \Bbb Z_3\Big\}$

To look for how many elements the multiplicative group has is not the same as to look for how many elements the general linear group has? Anyway, I know that with $\Bbb Z_n$ it is possible to use the Euler function to know how many elements the corresponding multiplicative group has. How should I proceed in this case? Any help?

  • $\begingroup$ Check if this group is abelian first. $\endgroup$ – Radost Mar 18 '19 at 16:57
  • $\begingroup$ @Radost Yes, It is $\endgroup$ – PCNF Mar 18 '19 at 16:58
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    $\begingroup$ Determine the number of matrices with nonzero determinant. $\endgroup$ – Wuestenfux Mar 18 '19 at 16:59
  • $\begingroup$ In this case you know that it's isomorphic to $\mathbb{Z}_n$ for some $n$. $\endgroup$ – Radost Mar 18 '19 at 16:59
  • $\begingroup$ @Radost how can I find $n$? $\endgroup$ – PCNF Mar 18 '19 at 17:06

There are only nine element of this ring, and the invertible ones satisfy $a^2+b^2\neq 0$. So write out the matrices and check the condition (You will find that there are 8 such matrices).

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  • $\begingroup$ I've done this before. Your answer, however, is useful to me in confirming that they are 8 invertible matrices. Thank you. $\endgroup$ – PCNF Mar 18 '19 at 18:29

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