Given a non-abelian finite simple group G of Lie type, and an arbitrary symmetric generating set S, it is conjectured by László Babai that its Cayley graph Γ(G, S) will have a diameter bound of (log |G|) O(1) . However, little progress has been made when the rank of G is large. In this article, we shall show that if G has rank n, and its base field has bounded size, then the diameter of Γ(G, S) would be bounded by exp (O(n(log n) 3 )).
AbstractATwo applications of our results are given, one in triangle patterns inside quasirandom groups and one in self-bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.