The Spread of Almost Simple Classical Groups

Abstract

Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every nontrivial element belongs to a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated. Which finite groups are $\frac{3}{2}$-generated? Every proper quotient of a $\frac{3}{2}$-generated group is cyclic, and in 2008, Breuer, Guralnick and Kantor made the striking conjecture that this condition alone provides a complete characterisation of the finite groups with this property. This conjecture has recently been reduced to the almost simple groups and results of Piccard (1939) and Woldar (1994) show that the conjecture is true for almost simple groups whose socles are alternating or sporadic groups. Therefore, the central focus is now on the almost simple groups of Lie type. In this monograph we prove a strong version of this conjecture for almost simple classical groups, building on earlier work of Burness and Guest (2013) and the author (2017). More precisely, we show that every relevant almost simple classical group has uniform spread at least two, unless it is isomorphic to the symmetric group of degree six. We also prove that the uniform spread of these groups tends to infinity if the size of the underlying field tends to infinity. To prove these results, we are guided by a probabilistic approach introduced by Guralnick and Kantor. This requires a detailed analysis of automorphisms, fixed point ratios and subgroup structure of almost simple classical groups, so the first half of this monograph is dedicated to these general topics. In particular, we give a general exposition of the useful technique of Shintani descent, which plays an important role throughout.