Citation
Landauer, Christopher Allen (1973) Simple Groups with 9, 10, and 11 Conjugate Classes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/V2KC3W98. https://resolver.caltech.edu/CaltechTHESIS:02012018102441931
Abstract
In this paper, we determine all simple groups with 9, 10 and 11 conjugate classes. The method we use is a modification of an old method of Landau: Suppose G is a finite group with n conjugate classes K_{1}, K_{2},...,K_{n}. Then the class equation for G can be written in the following form:
See PDF for formula
where m_{i} is the order of the centralizer of an element of K_{i}, and we choose the numbering so that G = m_{1}≥m_{2}≥···m_{n}. The method is to observe each solution and determine whether or not it corresponds to a simple group.
The main direction of this research was to develop tests that reduce the number of solutions computed. These tests deal primarily with the way various prime powers divide the m_{i}'s. These tests, together with a method for generating solutions to the class equation, were programmed by the author in FORTRAN for the IBM 370/155 at Caltech.
The computer time for the case n = 9 was 22 seconds, and for n = 10 it was about 7 minutes. For n = 11, the numbers involved were occasionally too large for the computer to deal with, and after producing several new tests, the computing time was 8 hours.
The effect of the computer programs was to produce a few hundred solutions of the class equation that it could not eliminate. These were then examined by hand in order to eliminate the ones that do not correspond to simple groups. During the eliminations by hand, new tests were discovered that should be mechanized for higher values of n.
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:  11 July 1972 
Record Number:  CaltechTHESIS:02012018102441931 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:02012018102441931 
DOI:  10.7907/V2KC3W98 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  10668 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  01 Feb 2018 23:55 
Last Modified:  17 Jul 2024 17:53 
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