TITLE: Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.
QUESTION [17 upvotes]: I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n \in \mathbb{N}\setminus\{0\}$. 
If $A$ is in the kernel, it is of the form $I + 3B$ where $B$ is an integer matrix. Beyond that, I don't have any ideas. None of the canonical forms seem to be helpful as these are integer valued matrices.
As always, any hints would be very much appreciated.

REPLY [2 votes]: Here's a proof that's basis free. It works with $3$ replaced by any odd prime. 
Let $A$ be a finite order element of the kernel. The first thing to realize is that $A$ has order a power of $3.$ To see this one observes that since $\langle A \rangle$ is finite and $\bigcap  \ker(SL_n(\mathbb{Z}) \rightarrow SL_n(\mathbb{Z}/3^d)) = 1,$ it must be the case $\langle A \rangle$ injects into $ \Gamma_d := \ker(SL_n(\mathbb{Z}/3^d) \rightarrow SL_n(\mathbb{Z}/3))$ for some large integer $d.$ Since  $ \Gamma_d$ is a $3$-group the claim follows. 
We now replace $A$ with some power and assume $A$ has order dividing 3.
Let $R$ be the ring $\mathbb{Z}_3[X]/(X^3 - 1)$ and $M$ be the $R$-module which is free over $\mathbb{Z}_3$ (the 3-adic integers) of rank $n$ where $X$ acts by $A.$ The ring $R$ is 1 dimensional. It has two minimal prime ideals: $$P_1 := (X -1) \text{ and } P_2 := (X^2 +X +1)$$ and one maximal ideal $\mathfrak{m} := (3, X - 1).$ One observes $$P_1 + P_2 = \mathfrak{m}.$$ To say $A =1$ is to say $R$ acts on $M$ through $R/P_1.$ This occurs if and only if $R/P_2$ acts on $R/P_2\otimes_R M $ through $R/P_1 + P_2 =R/\mathfrak{m}.$ 
The ring $R/P_2$ is a PID (in fact it's a DVR). Therefore by the classification of modules over a PID,  
$$M \otimes_R R/P_2 \cong \bigoplus_{i=1}^N  R/(\mathfrak{m}^{e_i} + P_2)$$
where $e_i$ is a positive number or infinity which depends on $i$.
Since $A$ is trivial modulo $3,$ one has $R$ acts on $M/3M$ through $R/(3)+ P_1 = R/\mathfrak{m}.$ 
But $R/((3) + \mathfrak{m}^{e_i} + P_2) \cong R/\mathfrak{m}$ only if $e_i = 1$. (This fact is not true if 3 is replaced by 2.) It follows $$R/P_2\otimes_R M \cong (R/\mathfrak{m})^N$$ and the claim follows.