Unitriangular-group diameter formulas

Prove that, for sufficiently large m depending on n, the diameters of U(n,Z/mZ) equal (n−1) floor(m/2) for fundamental undirected and positive undirected generators, equal (n−1)m+O(1) for fundamental oriented generators, and equal (n−1)m−2 for positive oriented generators.

Background

The paper studies unitriangular groups over Z/mZ with fundamental or positive-root transvections, in undirected and directed variants. Numerical experiments suggest that the diameter is eventually linear in the modulus m, improving the previously cited bounds.

References

For large enough $m$ (depending on $n$) the diameter of $U(n, \mathbb{Z}/m\mathbb{Z})$ is a follows:

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 9, subsection “Bounds on the diameters of unitriangular groups”