An improved diameter bound for finite simple groups of Lie type

© 2019 London Mathematical Society For a finite group (Formula presented.), let (Formula presented.) denote the maximum diameter of a connected Cayley graph of (Formula presented.). A well-known conjecture of Babai states that (Formula presented.) is bounded by (Formula presented.) in case (Formula presented.) is a non-abelian finite simple group. Let (Formula presented.) be a finite simple group of Lie type of Lie rank (Formula presented.) over the field (Formula presented.). Babai's conjecture has been verified in case (Formula presented.) is bounded, but it is wide open in case (Formula presented.) is unbounded. Recently, Biswas and Yang proved that (Formula presented.) is bounded by (Formula presented.). We show that in fact (Formula presented.) holds. Note that our bound is significantly smaller than the order of (Formula presented.) for (Formula presented.) large, even if (Formula presented.) is large. As an application, we show that more generally (Formula presented.) holds for any subgroup (Formula presented.) of (Formula presented.), where (Formula presented.) is a vector space of dimension (Formula presented.) defined over the field (Formula presented.)

A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank

We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This theorem generalizes an earlier theorem of the authors and Short. We also propose two conjectures that relate our result to one of Rodgers and Saxl pertaining to conjugacy classes in \SL_n(q), as well as to the Product Decomposition Conjecture of Liebeck, Nikolov and Shalev

Covering the Edges of a Connected Graph by Paths

AbstractWe prove that every connected graph onnvertices can be covered by at mostn/2+O(n3/4) paths. This implies that a weak version of a well-known conjecture of Gallai is asymptotically true

An extension of a frankl-füredi theorem

AbstractLet 1⩽r<n be integers and H a family of subsets of an n-element set such that 1⩽ |H∩H1|⩽r holds for all H, H1 ϵ H. Frankl and Füredi [3] proved that |H|⩽(n−10)+⋯+(n−1r) holds for n > 100r2log r and this is best possible. In this paper it is proved by using a new type of permutation method that the same holds for 6(r+1)⩽n⩽15(r+1)2

On the orders of doubly transitive permutation groups, elementary estimates

AbstractExtending ideas of L. Babai we give an nclog2n bound for the orders of 2-transitive groups of degree n not containing An. Our proof is elementary in the sense that it does not invoke the classification theorem of finite simple groups. We also give ideas leading to a short proof of a slightly weaker bound