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The rough question is:

How do the finite subgroups of $\operatorname{SL}(2,R/I)$ vary as $I$ varies, where $R$ is a nice local ring?

$$\operatorname{SL}(2,A) = \left.\left\{ \left(\begin{smallmatrix}a & b\\ c& d\end{smallmatrix}\right)\;\right\vert\;ad-bc=1, a,b,c,d \in A \right\}$$ is the group of a 2×2 matrices with entries from $A$ and determinant $1$.

Two cases seem basic and approachable but already very interesting:

  • $R=\mathbb{Z}_2$, the $2$-adic integers, and $R/I$ = $\mathbb{Z}/2^n\mathbb{Z}$ with $n$ varying, and
  • $R=\mathbb{F}_2[[x]]$, the power series over the field with $2$ elements, and $R/I = \mathbb{Z}[x]/(2,x^n)$ with $n$ varying.

In each case $G = \operatorname{SL}(2,R/I)$ is finite and solvable, with $G/O_2(G) ≅ \operatorname{SL}(2,2) ≅ S_3$ being a nice familiar group. In other words, nothing radically strange can happen in any one particular group.

However, I am having quite a bit of trouble understanding how the subgroups vary with $n$. Call the first sequence of groups $G(2,n)$ and the second sequence of groups $G(x,n)$.

Which groups $G(2,n)$ and $G(x,n)$ contain $Q_8$, a quaternion group of order $8$?

$Q_8$ seems to be contained in $G(x,2n+1)$ and not contained in $G(x,2n)$ or $G(x,1)$ or any $G(2,n)$.

Which dihedral groups are contained in each $G(2,n)$ and $G(x,n)$?

The $G(2,n+1)$ seem to be bereft of dihedrals, but $G(2,1)$ has one, and $G(x,n+1)$ seem to have the full crew. Why does the "$2$" destroy dihedrals while the "$x$" invites more? On the abelian side:

What is the $2$-rank (maximum $n$ such that $G$ contains the direct product of $n$ cyclic groups of order $2$) of each $G(2,n)$ and $G(x,n)$?

Again the pattern seems a bit wobbly, especially if you count how many subgroups of each rank there are (these do not form monotonic sequences).

Any answer to the bold question should be able to address the strange (conjectured) answers to the smaller questions, but I'm also just fine with some ad hoc answers to the small ones as these groups are currently a complete mystery. I think larger primes (or residue fields) would help with the very restricted nature of the subgroups available for testing, but at the beginning $p=2$ seems to help. More independent nilpotents has not struck me as counter-intuitive yet. Also in my limited experiments, $\operatorname{GL}$ seems to be a lot less strange, but feel free to share observations about it as well.

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There are quite a few recent questions about SL(2,R) for arbitrary commutative R on The most promising techniques reduce to artinian local rings, like the ones I mention here. However, the associated groups do not agree with my intuition at all. – Jack Schmidt Dec 5 '10 at 21:23

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