, Volume 99, Issue 5, pp 417-424
Date: 17 Nov 2012

Diameters of Chevalley groups over local rings

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Abstract

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 .