Simple Groups with 9, 10, and 11 Conjugate Classes

Citation

Landauer, Christopher Allen (1973) Simple Groups with 9, 10, and 11 Conjugate Classes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/V2KC-3W98. https://resolver.caltech.edu/CaltechTHESIS:02012018-102441931

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 K₁, K₂,...,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| = m₁≥m₂≥···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.

Thesis Files