Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the surjective map $\phi_p : \text{GL}_n(\mathbb{Z}) \rightarrow \text{GL}_n(\mathbb{Z}/p)$ is ``injective on the torsion''. In other words, if $F$ is a finite subgroup of $\text{GL}_n(\mathbb{Z})$, then $\phi_p|_F$ is injective.

Fixing an odd prime $p$, I have two questions about the map $\phi_p$.

1. Are there any interesting restrictions on elements $y \in \text{GL}_n(\mathbb{Z}/p)$ such that there exists some finite-order $x \in \text{GL}_n(\mathbb{Z})$ with $\phi_p(x) = y$?

2. Do there exist any non-conjugate finite order elements $x,x' \in \text{GL}_n(\mathbb{Z})$ such that $\phi_p(x) = \phi_p(x')$? The hypothesis that $x$ and $x'$ are non-conjugate is to rule out the silly example where $x' = g x g^{-1}$ with $g \in \text{ker}(\phi_p)$.

Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map $\phi_p : \text{GL}_n(\mathbb{Z}) \rightarrow \text{GL}_n(\mathbb{Z}/p)$ is ``injective on the torsion''. In other words, if $F$ is a finite subgroup of $\text{GL}_n(\mathbb{Z})$, then $\phi_p|_F$ is injective.

Fixing an odd prime $p$, I have two questions about the map $\phi_p$.

1. Are there any interesting restrictions on elements $y \in \text{GL}_n(\mathbb{Z}/p)$ such that there exists some finite-order $x \in \text{GL}_n(\mathbb{Z})$ with $\phi_p(x) = y$?

2. Do there exist any non-conjugate finite order elements $x,x' \in \text{GL}_n(\mathbb{Z})$ such that $\phi_p(x) = \phi_p(x')$? The hypothesis that $x$ and $x'$ are non-conjugate is to rule out the silly example where $x' = g x g^{-1}$ with $g \in \text{ker}(\phi_p)$.

Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the surjective map $\phi_p : \text{GL}_n(\mathbb{Z}) \rightarrow \text{GL}_n(\mathbb{Z}/p)$ is ``injective on the torsion''. In other words, if $F$ is a finite subgroup of $\text{GL}_n(\mathbb{Z})$, then $\phi_p|_F$ is injective.

Fixing an odd prime $p$, I have two questions about the map $\phi_p$.

1. Are there any interesting restrictions on elements $y \in \text{GL}_n(\mathbb{Z}/p)$ such that there exists some finite-order $x \in \text{GL}_n(\mathbb{Z})$ with $\phi_p(x) = y$?

2. Do there exist any non-conjugate finite order elements $x,x' \in \text{GL}_n(\mathbb{Z})$ such that $\phi_p(x) = \phi_p(x')$?

Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the surjective map $\phi_p : \text{GL}_n(\mathbb{Z}) \rightarrow \text{GL}_n(\mathbb{Z}/p)$ is ``injective on the torsion''. In other words, if $F$ is a finite subgroup of $\text{GL}_n(\mathbb{Z})$, then $\phi_p|_F$ is injective.

Fixing an odd prime $p$, I have two questions about the map $\phi_p$.

1. Are there any interesting restrictions on elements $y \in \text{GL}_n(\mathbb{Z}/p)$ such that there exists some finite-order $x \in \text{GL}_n(\mathbb{Z})$ with $\phi_p(x) = y$?

2. Do there exist any non-conjugate finite order elements $x,x' \in \text{GL}_n(\mathbb{Z})$ such that $\phi_p(x) = \phi_p(x')$? The hypothesis that $x$ and $x'$ are non-conjugate is to rule out the silly example where $x' = g x g^{-1}$ with $g \in \text{ker}(\phi_p)$.