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Hurwitz groups and \(G_ 2(q)\). (English) Zbl 0734.20013
Summary: Finite factor groups of \(G_{2,3,7}:=\langle\sigma_ 1,\sigma_ 2,\sigma_ 3\mid\sigma^ 2_ 1=\sigma^ 3_ 2=\sigma^ 7_ 3=\sigma_ 1\sigma_ 2\sigma_ 3=1\rangle\) are called Hurwitz groups. Here we prove that apart from \(^ 2G_ 2(3)\), \(G_ 2(2)\), \(G_ 2(3)\) and \(G_ 2(4)\), all the groups \(^ 2G_ 2(3^{2n+1})\) and \(G_ 2(q)\), \(q=p^ n\), \(p\in{\mathbb{P}}\), are Hurwitz groups. For the proof, \((2,3,7)\) structure contants are calculated from the character tables [B. Chang, R. Ree, Symp. Math. 13, 395-413 (1974; Zbl 0314.20034); H. Enomoto, Jap. J. Math., New Ser. 2, 191-248 (1976; Zbl 0384.20007)], and then with the lists of maximal subgroups [P. Kleidman, J. Algebra 117, 30-71 (1988; Zbl 0651.20020); B. Cooperstein, ibid. 70, 23-36 (1981; Zbl 0459.20007)] the existence of generating triples is deduced.

20F05 Generators, relations, and presentations of groups
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20D05 Finite simple groups and their classification
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20D06 Simple groups: alternating groups and groups of Lie type
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