Abstract

Let $G$ be a finite group. A generating set of $G$ is called symmetric if it contains the inverses of its elements. The diameter of a generating set $S$ of $G$ is the smallest positive integer $n$ such that every element of $G$ is a product of at most $n$ elements of $S$. The diameter of $G$ is the maximum diameter of a symmetric generating set of $G$. In other words, the diameter of $G$ is the maximum diameter of a connected Cayley graph of $G$. In 1992, László Babai brilliantly conjectured that the diameter of any finite simple group $G$ is bounded above by $(\log|G|)^c$, where $c$ is an absolute constant. L. Pyber’s program to prove this conjecture is to first reduce the problem to generating sets containing elements with small support; for this reason, it makes sense to solve the conjecture for such generating sets. In a joint work with Zoltán Halasi and Gábor Somlai, we considered symplectic, unitary and special linear simple groups and we proved the conjecture for such groups in the special case in which the generating set $S$ contains a transvection. In this talk I will present our result and give the main ideas of its proof.

Joint with Zoltán Halasi and Gábor Somlai.