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Pairs of Generators of the Known Simple Groups whose Orders are Less than One Million

1930; Princeton University; Volume: 31; Issue: 4 Linguagem: Inglês

10.2307/1968152

1939-8980

H. R. Brahana,

Advanced Topics in Algebra

A group of order k *n which admits of a graphical representation by means of k n-sided regions on a two-sided surface may be generated by an operator of order two and an operator of order n; and, conversely, any group of order k. n which may be generated by an operator of order two and an operator of order n may be so represented Among the groups so determined appear two infinite systems of simple groups, the alternating groups of degree greater than 4, and the subgroups modulo p of the modular group. It is perhaps to be expected that every simple group of composite order, or at least every simple group of even order, can be generated by two operators of which one is of order two. In the absence of any proof of the conjectured theorem it seemed expedient to examine some special cases with a view to its contradiction or the accumulation of facts bearing on the range of its validity. Accordingly we examined the known simple groups whose orders are less than one million and herewith present a pair of generators of the type described for each such group. With five noteworthy exceptions, viz.; the multiply transitive groups of Mathieu, every known simple group of composite order belongs to one or more of nine infinite classes. Of these nine classes eight are represented among the 53 known simple groups of order less than one million,4 and, moreover, 3 of the 5 Mathieu groups are also present. The class of simple groups FO (2 m, pn) has no member among these groups, the lowest order of such a group being 3, 265, 920. In order to have our investigations include at least one member of each of the infinite classes we included this group. The Mathieu group on 23 letters was included because many of the necessary facts came to light in the search for generators of its simple subgroup on 22 letters. Thirty-two of the fifty-three known simple groups of order less than a million belong to the two infinite classes described above and hence are known to be generated by two operators of which one is of order two. For each of the remaining groups we select two operators S and T in the