AbstractIt is well known that every ﬁnite simple group can be generated by two elements. 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 3/2-generated.
It is natural to ask which groups are 3/2-generated. It is easy to see that every proper quotient of a 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. That is, they conjectured that a ﬁnite group is 3/2-generated if and only if every proper quotient of the group is cyclic. This conjecture has been reduced to the almost simple groups through recent work of Guralnick. By work of Piccard (1939) and Woldar (1994), the conjecture is known to be true for almost simple groups whose socles are alternating or sporadic groups. Therefore, the central focus is the almost simple groups of Lie type.
In this thesis we prove a stronger version of this conjecture for almost simple symplectic and orthogonal groups, building on earlier work of Burness and Guest (2013) for linear groups. More generally, we study the uniform spread of these groups, obtaining lower bounds and related asymptotics. We adopt a probabilistic approach using ﬁxed point ratios, which relies on a detailed analysis of the conjugacy classes and subgroup structure of the almost simple classical groups.
|Date of Award||25 Jun 2019|
|Supervisor||Tim Burness (Supervisor)|