On the Spread of Classical Groups

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

Abstract

It 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 Award25 Jun 2019
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorTim C Burness (Supervisor)

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