# On the coefficients that appear in finite groups of matrices with integer entries

Let $$n$$ be a positive integer and $$G$$ be a finite group of $$n\times n$$ matrices with integer coefficients, i.e. $$G\subset\operatorname{GL}_n(\mathbb{Z})$$. It is known that for sufficiently large $$n$$, the maximum order of such a group is $$2^nn!$$ by Feit (although relying on an unpublished manuscript of Weisfeiler I believe) and it is attained. I am interested in the coefficients that appear in the matrices of such a group. Specifically, I would like to know how many distinct numbers can appear.

Clearly it cannot be more than $$2^nn!n^2$$ if all coefficients of all matrices are distinct but I have the feeling that such a bound cannot be reached. By considering the companion matrix of a cyclotomic polynomial, I think it is possible to produce roughly $$n$$ or maybe even $$n^2$$ distinct coefficients. On the other hand, a group like the symmetric group can be encoded with $$0,1$$ matrices and the group that attains the $$2^nn!$$ upper bound only uses $$-1,0,1$$.

My questions are:

• is it possible to derive a much better bound on the number of distinct coefficients, for example polynomial in $$n$$ ?

or

• is there an example family of finite groups whose number of distinct coefficients is exponential in $$n$$ ?

I have the feeling that representation theory could help but I am not well-versed in the theory of finite groups.

It is possible to get a number of distinct coefficients exponential in $$n$$. Here is an example.
Let $$B = \begin{bmatrix} 1 & -1 & -1 & -1 & \cdots & x_1 \\ 0 & 1 & 0 & 0 & & x_2 \\ 0 & 0 & 1 & 0 & \cdots & x_3 \\ 0 & 0 & 0 & 1 & & \vdots \\ & \vdots & & & \ddots & x_{n-1} \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$$ We have $$B^{-1} = \begin{bmatrix} 1 & 1 & 1 & 1 & \cdots & -S \\ 0 & 1 & 0 & 0 & & -x_2 \\ 0 & 0 & 1 & 0 & \cdots & -x_3 \\ 0 & 0 & 0 & 1 & & \vdots \\ & \vdots & & & \ddots & -x_{n-1} \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$$ where $$S=x_1+x_2+\cdots+x_{n-1}$$.
We consider the conjugate by $$B$$ of the group of diagonal $$\pm 1$$ matrices, of size $$2^n$$. Let $$X$$ be a diagonal matrix with $$X_{i,i}=\varepsilon_i \in \{\pm 1\}$$ and suppose $$\varepsilon_n=1$$. Then we can compute the upper right entry of $$B^{-1}XB$$ as $$-S + \sum_{i=1}^{n-1} \varepsilon_i x_i$$ If we take, for example, $$x_i=2^i$$, then these sums are all different for the $$2^{n-1}$$ possible choices of $$X$$, so at least $$2^{n-1}$$ different coefficients appear in the matrices of the group.