# abelian subgroups of $GL(2,\mathbb{Z}_p)$

Is there a classification of abelian subgroups of $GL(2,\mathbb{C})$? or $GL(2,\mathbb{Z}_p)$?

Here $\mathbb{Z}_p$ is the ring of $p$-adic integers.

• For abelian subgroups of $GL(2,p)$ see here. – Dietrich Burde Aug 21 '15 at 19:12
• For $\Bbb{Z}_p$ there are other abelian subgroups. For example if $K$ is the quadratic unramified extension of $\Bbb{Q}_p$, then its ring of integers $\mathcal{O}$ is a free $\Bbb{Z}_p$ module of rank two, and its group of units $\mathcal{O}^*$ is then surely acting faithfully on it, and can thus be embedded into $GL_2(\Bbb{Z}_p)$. – Jyrki Lahtonen Aug 21 '15 at 19:18
• Do you denote by $\mathbb{Z}_p$ the $p$-adic integers, or the finite field $\mathbb{F}_p$ ? The notation $GL(2,p)$ refers to the finite field. – Dietrich Burde Aug 21 '15 at 19:35
• @DietrichBurde I think he means $\mathbb Z/p\mathbb Z$. – principal-ideal-domain Aug 21 '15 at 19:36
• Regarding finite abelian subgroups of $GL(2,\mathbb{C})$ there is a classification, see here. – Dietrich Burde Aug 21 '15 at 19:45