# How many elements has the multiplicative group of this ring?

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?

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