# Centralizers of Cartan subgroup III

Let $$\mathcal O$$ be an order in an imaginary quadratic field $$K$$. Let $$n$$ be a positive integer. The multiplicative group $$(\mathcal O/n\mathcal O)^\times$$ acts on the module $$\mathcal O/n\mathcal O\cong \mathbb Z / n\mathbb Z\times \mathbb Z / n\mathbb Z$$. We get thus an embedding of $$(\mathcal O/n\mathcal O)^\times$$ into $$\operatorname{GL}_2(\mathbb Z / n\mathbb Z)$$. Let $$C_n$$ be the image.

Is the centralizer of $$C_n$$ in $$\operatorname{GL}_2(\mathbb Z / n\mathbb Z)$$ equal to $$C_n$$ itself?

I am interested especially in the case when $$n$$ is a power of two.

• What is the difference to your previous question mathoverflow.net/questions/327168/… ? – Chris Wuthrich Apr 20 '19 at 23:12
• @ChrisWuthrich, I still do not know how to prove this if $n$ is a power of two. – Shimrod Apr 21 '19 at 6:14