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Permutation representations of group quotients and of quasisimple groups
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Chamberlain, Robert (2020) Permutation representations of group quotients and of quasisimple groups. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3493077~S15
Abstract
The minimal degree, µ(G), of a finite group G is the least n such that G embeds in Sn. Such embeddings, called permutation representations, are often used to represent groups on computers. Algorithms working with such representations have time and space complexity depending on n so it is often worth putting some time into getting n as close to µ(G) as possible.
In the second chapter of this thesis we study group quotients. Despite a quotient G/N of G being smaller and in some sense simpler than G it possible to have µ(G/N) > µ(G), in which case G is called exceptional and N is called distinguished in G. We characterise exceptional pgroups of least order and show that normal subgroups with no abelian chief factors are not distinguished. These develop from work by Kovács, Easdown and Praeger.
In the third chapter we study quasisimple groups. The most significant result in the third chapter is the calculation of µ(2·An) for all n. This is in some sense the worst case for the minimal degree of a quasisimple group as µ(2 · An) grows with (n/2)!. A representation of degree µ(2 · An) is first given, then the proof that it is minimal comes in two parts. We describe a dynamic programming algorithm for computing µ(2 · An) for small n. This is done for n ≤ 850. For n > 850 we use an inductive proof to compute µ(2 · An).
We also compute µ(SL(n, q)) following work by Cooperstein and conclude with comments on the minimal degrees of other classical groups and of Schur covers of some sporadic simple groups.
Item Type:  Thesis (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Group Theory, Permutation groups  
Official Date:  April 2020  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Format of File:  
Extent:  120 leaves  
Language:  eng 
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