Citation
Sibley, David Alan (1972) On certain finite linear groups of prime degree. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CW3F2D77. https://resolver.caltech.edu/CaltechTHESIS:06132016162549466
Abstract
In studying finite linear groups of fixed degree over the complex field, it is convenient to restrict attention to irreducible, unimodular, and quasiprimitive groups. If one assumes the degree to be an odd prime p, there is a natural division into cases, according to the order of a Sylow pgroup of such a group. When the order is p^{4} or larger, all such groups are known (by W. Feit and J. Lindsey, independently).
THEOREM 1. Suppose G is a finite group with a faithful, irreducible, unimodular, and quasiprimitive complex representation of prime degree p ≥ 5. If a Sylow pgroup P of G has order p^{3}, then P is normal in G.
As is well known, Theorem 1 is false for p = 2 or 3. Combining Theorem 1 with known results, we have immediately the following conjecture of Feit.
THEOREM 2. Suppose G is a finite group with a faithful, irreducible, and unimodular complex representation of prime degree p ≥ 5. Then p^{2} does not divide the order of G/0_{p}(G).
The following result, which is of independent interest, is used in the proof of Theorem 1.
THEOREM 3. Suppose G is a finite group with a Sylow pgroup P of order larger than 3, which satisfies
C_{G}(x) = P, for all x ≠ 1 in P.
If G has a faithful complex representation of degree less than (P  1)^{2/3}, then P is normal in G.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  3 April 1972  
Funders: 
 
Record Number:  CaltechTHESIS:06132016162549466  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:06132016162549466  
DOI:  10.7907/CW3F2D77  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9874  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  14 Jun 2016 15:21  
Last Modified:  09 Nov 2022 19:20 
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