$\newcommand{\GL}{\mathrm{GL}}\newcommand{\dbZ}{\mathbb{Z}}\newcommand{\dbF}{\mathbb{F}}\newcommand{\dbN}{\mathbb{N}}$ Hello.
I have a small question regarding closed subgroup of $\GL_n(\dbZ_p)$, which I tried floating on MSE but did not get a response. As this question arose from my research I figured it might be suitable to ask it here.
Fix $n\in \dbN$ and let $p$ be a fixed prime. Let $\eta:\GL_n(\dbZ_p)\to\GL_n(\dbF_p)$ denote the coordinate-wise reduction-map.
Suppose we have two closed (with respect to $p$-adic topology) and infinite subgroups $G, H\in\GL_n(\dbZ_p)$, which have the same image under $\eta$, i.e. $\eta(G)=\eta(H)$.
What can said in this case about $G$ and $H$? I know one can not expect them to be equal (e.g. $G=1+p M_n(\dbZ_p)$ and $H=1+p^k M_n(\dbZ_p)$ have the same image mod $p$).
Can it be proved for example that in such a case $G$ and $H$ are commensurable? perhaps under some additional assumptions regarding these groups? $\newcommand{\GG}{\mathsf{G}}\newcommand{\HH}{\mathsf{H}}$
The main focus of interest is for the case where $G$ and $H$ arise as the group of $\dbZ_p$ points of (smooth) algebraic groups $\GG,\HH$, defined over $\dbZ_p$.
I remark in this case there is no assumption that either group has good reduction modulo $p$ and that I am intentionally looking at coorindate-wise reduction (for which I don't know of any functorial interpretation), rather than the better behaved operation of reduction mod $p$ of the group schemes $\GG$ and $\HH$ (i.e. the $\dbF_p$-points of the image of the functor $\newcommand{\spec}{\mathrm{Spec}} \GG\mapsto \GG\times_{\spec\dbZ_p}\spec\dbF_p$, and similarly for $\HH$).
I would very much appreciate any sort of idea or refernece anyone might have to offer on the subject.
Thank you.
