On conjectures of Guralnick and Thompson (Q2644754)

If s is a permutation on a set \(\Omega\) of order n, then c(s) denotes the number of cycles of s and \(Ind(s)=n-c(s)\). By definition, a genus g system is a triple (G,\(\Omega\),S) where \(\Omega\) is a finite set, G is a transitive subgroup of Sym(\(\Omega\)) and \(S=\{g_ j|\) \(1\leq j\leq r\}\) is an ordered subset of \(G^{\#}\) such that \(G=<S>\), \(g_ 1g_ 2...g_ r=1\) and \(2(| \Omega | +g-1)=\sum^{r}_{j=1}Ind(g_ j).\) This condition is equivalent to the existence of a branched covering of the Riemann sphere by a Riemann surface of genus g with monodromy group G. Thompson conjectured that if (G,\(\Omega\),S) is a genus 0 system then the composition factors of G are, with a finite number of exceptions, of prime order or alternating groups. Also Thompson has observed that the conjecture reduces to the case where G is primitive on \(\Omega\). Guralnick has suggested a stronger conjecture: There exist constants \(C_ 1\), \(C_ 2\) such that if (G,\(\Omega\),S) is a primitive genus g system, then either \(n\leq C_ 1g+C_ 2\) or the composition factors of G are of prime order or alternating groups. Primitive groups have been shown to fall into one of five classes [in [1], \textit{M. Aschbacher} and \textit{L. Scott}, J. Algebra 92, 44-80 (1985; Zbl 0549.20011)]. The main result of this paper is: Theorem. Let (G,\(\Omega\),S) be a genus g system such that 336(g-1)\(\leq n\) and such that (G,\(\Omega\)) is primitive and belongs to the class C2 of [1]. Then \(g=0\) or 1, G is of index 2 or 4 in \(S_ 5 wr Z_ 2\). Also up to equivalence there are 8 systems (G,\(\Omega\),S) with \(g=0\) and 11 systems (G,\(\Omega\),S) with \(g=1\). The notion of equivalence is given in Section 4 and is essentially the composition of a permutation isomorphism with an element of the braid group. In particular, the Thompson and Guralnick conjectures hold in case C2 of [1, Theorem].