Diameter of Cayley graphs of SL(n,p) with generating sets containing a transvection

A well-known conjecture of Babai states that if G is a finite simple group and X is a generating set of G, then the diameter of the Cayley graph Cay ( G , X ) is bounded above by ( log ⁡ | G | ) c for some absolute constant c. The goal of this paper is to prove such a bound for the diameter of Cay ( G , X ) whenever G = S L ( n , p ) and X is a generating set of G which contains a transvection. A natural analogue of this result is also proved for G = S L ( n , K ) , where K can be any field.