Bounds of fixed point ratios of permutation representations of GL_n(q) and groups of genus zero

Author: Shih, Tanchu

Year: 1991

Degree: Dissertation (Ph.D.)

Advisor: Aschbacher, Michael

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/a3e6-tj54

Abstract

If G is a transitive subgroup of the symmetric group Sym (Ω), where Ω is a finite set of order m; and G satisfies the following conditions: G=, S={g_1,…,g_r] ⊆ G^#, g_1…g_r=1, and r∑i=1 c(g_i)=(r-2)m+2, where c(g_i) is the number of cycles of g_1 on Ω, then G is called a group of genus zero. These conditions correspond to the existence of an m-sheeted branched covering of the Riemann surface of genus zero with r branch points. The fixed point ratio of an element g in G is defined as f(g)/|Ω|, where f(g) is the number of fixed points of g on Ω. In this thesis we assume that G satisfies L_n(q) ≤G≤PGL_n(q) and G is represented primitively on Ω. The primitive permutation representations of G are determined by the maximal subgroups of G. The bounds are expressed as rational functions which depend on n, q, the rational canonical forms of the elements, and the maximal subgroups. Then those bounds are used to prove the following: Theorem: If G is a group of genus zero, then one of the following holds: (a) q=2 and n≤32, (b) q=3 and n≤12, (c) q=4 and n≤11, (d) 5≤q≤13 and n≤8, (e) 16≤q≤83 and n≤4, (f) 89≤q≤343 and n=2. Thus for those G satisfying L_n(q) ≤G≤PGLn(q), this theorem confirms the J. Thompson’s conjecture which states that except for Z_p, A_k with k≥5, there are only finitely many finite simple groups which are composition factors of groups of genus zero.