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Sporadic simple groups which are Hurwitz. (English) Zbl 0736.20012
In an earlier paper [Ill. J. Math. 33, 416-437 (1989; Zbl 0654.20014)] the author determined for 19 of the 26 sporadic simple groups whether or not they are Hurwitz groups, i.e. generated by elements \(x\) and \(y\) of orders 2 and 3 whose product has order 7. Three more were dealt with by the reviewer with Kleidman, Linton and Parker. In the present paper the author considers the cases \(J_ 4\) and \(Fi_{22}\), proving that they are both Hurwitz groups. The proof depends crucially on the lists of maximal subgroups computed by Kleidman and Wilson, and proceeds by an exhaustive analysis of the structure constants of type (2,3,7). Recently the reviewer has shown (unpublished) that \(B\) is not a Hurwitz group, which leaves only the case of the Monster undecided.

MSC:
20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups
20E28 Maximal subgroups
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