Archiv der Mathematik

, Volume 99, Issue 5, pp 417–424 | Cite as

Diameters of Chevalley groups over local rings

  • Oren Dinai


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 .


Local Ring Cayley Graph Chevalley Group Chevalley Basis International Mathematics Research Notice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsETH Zurich Ramistrasse 101ZurichSwitzerland

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