Generation of finite simple groups
File(s)
Author(s)
King, Carlisle
Type
Thesis
Abstract
Let G be a finite simple group. Well-known results of Miller, Steinberg and Aschbacher-Guralnick prove that G can be generated by a pair of elements - we say that G is 2-generated. In this thesis, we consider some variations of this result.
A natural refinement of the 2-generation result is to ask, for a pair of integers (a,b), whether a finite simple group G is generated by an element of order a and an element of order b. The smallest pair of interest is (2,3) (pairs of elements of order 2 generate dihedral groups). Work of Liebeck-Shalev and Lubeck-Malle shows that, apart from a few known infinite families, all but finitely many finite simple groups are (2,3)-generated. The first major result of this thesis proves that every finite simple group can be generated by an element of order 2 and an element of prime order.
An equivalent statement of the 2-generation theorem is that every finite simple group is an image of F_2, the free group on 2 generators. More generally, given a finitely presented group H, one can ask which finite simple groups are images of H. A result of Liebeck and Shalev shows that given a pair (A,B) of nontrivial finite groups such that at least one of A or B is not a 2-group, every finite simple classical group of sufficiently large rank is an image of A star B, the free product of A and B. The second major result of this thesis generalizes this result by proving the same conclusion holds for pairs (A,B) of any nontrivial finite groups, not both of order 2.
A natural refinement of the 2-generation result is to ask, for a pair of integers (a,b), whether a finite simple group G is generated by an element of order a and an element of order b. The smallest pair of interest is (2,3) (pairs of elements of order 2 generate dihedral groups). Work of Liebeck-Shalev and Lubeck-Malle shows that, apart from a few known infinite families, all but finitely many finite simple groups are (2,3)-generated. The first major result of this thesis proves that every finite simple group can be generated by an element of order 2 and an element of prime order.
An equivalent statement of the 2-generation theorem is that every finite simple group is an image of F_2, the free group on 2 generators. More generally, given a finitely presented group H, one can ask which finite simple groups are images of H. A result of Liebeck and Shalev shows that given a pair (A,B) of nontrivial finite groups such that at least one of A or B is not a 2-group, every finite simple classical group of sufficiently large rank is an image of A star B, the free product of A and B. The second major result of this thesis generalizes this result by proving the same conclusion holds for pairs (A,B) of any nontrivial finite groups, not both of order 2.
Version
Open Access
Date Issued
2018-05
Date Awarded
2018-09
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Liebeck, Martin
Sponsor
Engineering and Physical Sciences Research Council
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)