, Volume 99, Issue 5, pp 417-424

Diameters of Chevalley groups over local rings

Purchase on Springer.com

$39.95 / €34.95 / £29.95*

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


Let G be a Chevalley group scheme of rank l. Let ${G_n := G(\mathbb{Z} / p^{n} \mathbb{Z})}$ be the family of finite groups for ${n \in \mathbb{N}}$ and some fixed prime number pp 0. We prove a uniform poly-logarithmic diameter bound of the Cayley graphs of G n with respect to arbitrary sets of generators. In other words, for any subset S which generates G n , any element of G n is a product of C n d elements from ${S \cup S^{-1}}$ . Our proof is elementary and effective, in the sense that the constant d and the functions p 0(l) and C(l, p) are calculated explicitly. Moreover, we give an efficient algorithm for computing a short path between any two vertices in any Cayley graph of the groups G n .