Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$. For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be the smallest $k \in \mathbb{N}$ such that $\bigcup_{i=0}^k \mathcal{M}^i = G$, i.e., every matrix in $G$ can be written as a product of matrices from $\mathcal{M}$, with product length at most $k$. Further, define the diameter $\mathit{diam}(G)$ of $G$ as the maximum $\mathcal{M}$-diameter of $G$, over all sets $\mathcal{M}$ generating $G$. Define the function $d : \mathbb{N} \to \mathbb{N}$ such that $d(n)$ is the maximum $\mathit{diam}(G)$, over all finite subgroups $G$ of $\mathrm{GL}(n,\mathbb{Q})$.

Equivalently, $d(n)$ can be formally defined on one line by $$ d(n) \ := \ \max \left\{\min\left\{k \in \mathbb{N} : \bigcup_{i=0}^k \mathcal{M}^i = \langle\mathcal{\mathcal{M}}\rangle\right\} : \langle\mathcal{\mathcal{M}}\rangle \le \mathrm{GL}(n,\mathbb{Q}) \text{ is finite}\right\}\,. $$

I am interested in lower and upper bounds on $d(n)$.

It is known from [S. Friedland. The maximal orders of finite subgroups in GL$_n$($\mathbb{Q}$). Proceedings of the American Mathematical Society, 125(12):3519-3526, 1997] that for large $n$ the maximal order of a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$ is $2^n n!$. Hence, $d(n) \le 2^{C n \log n}$ for some $C>0$.

On the other hand, denoting by $p(n)$ the maximal order of a permutation on $n$ elements, we have $\lim_{n \to \infty} (\ln p(n))/{\sqrt{n \ln n}} = 1$; see, e.g., [W. Miller. The maximum order of an element of a finite symmetric group. American Mathematical Monthly, 94(6):497-506, 1987]. Hence, $d(n) \ge 2^{c \sqrt{n \log n}}$ for some $c>0$.

Are there better bounds on $d(n)$?


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