RANDOM GENERATION OF SIMPLE GROUPS BY TWO CONJUGATE ELEMENTS

Abstract

Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as [mid ]G[mid ]→∞. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability P_c(G) that 〈x, x^y〉=G, where x, y∈G are chosen at random (with uniform distribution on G×G). We shall show that P_c(G)→1 as [mid ]G[mid ]→∞ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with.