Simple Groups with 9, 10, and 11 Conjugate Classes

Author: Landauer, Christopher Allen

Year: 1973

Degree: Dissertation (Ph.D.)

Advisor: Wales, David B.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/V2KC-3W98

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,...,Kn. Then the class equation for G can be written in the following form:

See PDF for formula

where mi is the order of the centralizer of an element of Ki, and we choose the numbering so that |G| = m1≥m2≥···mn. 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 mi'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.

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