In this paper, we determine all simple groups with 9, 10 and11 conjugate classes. The method we use is a modification of anold method of Landau: Suppose G is a finite group with nconjugate classes K1, K2,...,Kn. Then the class equation for Gcan be written in the following form:
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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 notit corresponds to a simple group.
The main direction of this research was to develop tests thatreduce the number of solutions computed. These tests deal primarilywith 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 370/155at Caltech.
The computer time for the case n = 9 was 22 seconds, and forn = 10 it was about 7 minutes. For n = 11, the numbers involved were occasionally too large for the computer to deal with, and afterproducing 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 onesthat do not correspond to simple groups. During the eliminationsby hand, new tests were discovered that should be mechanized for higher values of n.
