Let G be a Chevalley group scheme of rank l. Let G _n:= G(□/p ~n□) be the family of finite groups for n ∈ ? and some fixed prime number p > p _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 Cn ~d elements from S ∪ 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.
展开▼
