Solutions of a matrix equation over $\mathbb{Z}$

Question

Does the equation

$$ X^n-I=0 $$

have solution in $GL_4(\mathbb{Z})$ different from $I$, for every $n\geq 2$? If yes, how can I found them?

Answer 1

No, this has not always a solution. For example, for every prime $p\ge 7$, the equation $A^p=I$ has no solution in $GL_4(\mathbb{Z})$. Recall that the order of an element $A\in GL_n(\mathbb{Z})$ is the minimal $k\ge 1$ such that $A^k=I$. Now the maximal order $G(n)$ of an element $A\in GL_n(\mathbb{Z})$ satisfies the asymtotic relation $$ \log (G(n))\sim \sqrt{n\log(n)}. $$ Furthermore, let $L\colon \mathbb{N}^{\ast}\rightarrow \mathbb{N}$ be the additive function defined by $L(1)=L(2)=0$, and $$ L(p^r)=p^r-p^{r-1},\quad \text{if } p^r\ge 3. $$ The we have the following result:

Proposition: An integer $k$ is the order of an element $A\in GL_n(\mathbb{Z})$ if and only if $L(k)\le n$.

For the easy proof, see Proposition $1.1$ here. For $n=4$ and $k=p>5$ we have $L(p)=p-1>4=n$.