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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 Recently the reviewer has shown (unpublished) that $$B$$ is not a Hurwitz group, which leaves only the case of the Monster undecided.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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References:
 [1] Conway, J.H; Curtis, R.T; Norton, S.P; Parker, R.A; Wilson, R.A, () [2] Isaacs, I.M, Character theory of finite groups, (1976), Academic Press New York · Zbl 0337.20005 [3] Janko, Z, A new finite simple group of order 86, 775, 571, 046, 077, 562, 880 which possesses M24 and the full covering group of M22 as subgroups, J. algebra, 42, 564-596, (1976) [4] Kleidman, P.B; Parker, R.A; Wilson, R.A, The maximal subgroups of the fischer group fi23, J. London math. soc., 39, 89-101, (1989), (2) · Zbl 0629.20006 [5] Kleidman, P.B; Wilson, R.A, The maximal subgroups of J4, (), 484-510, (3) · Zbl 0619.20004 [6] Kleidman, P.B; Wilson, R.A, The maximal subgroups of fi22, (), 17-23, No. 17 · Zbl 0622.20008 [7] Serre, J.-P, Linear representations of finite groups, (1977), Springer-Verlag New York [8] Wilson, R.A, On maximal subgroups of the fischer group fi22, (), 197-222 · Zbl 0551.20010 [9] {\scS. A. Linton and R. A. Wilson}, private communication. [10] Woldar, A.J, On Hurwitz generation and genus actions of sporadic groups, Illinois J. math., 33, No. 3, 416-437, (1989) · Zbl 0654.20014 [11] Woldar, A.J, Representing M11, M12, M22 and M23 on surfaces of least genus, Comm. algebra, 18, No. 1, 15-86, (1990)
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