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


Shih, Tanchu (1991) Bounds of fixed point ratios of permutation representations of GL_n(q) and groups of genus zero. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/a3e6-tj54.


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>, 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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Aschbacher, Michael
Thesis Committee:
  • Unknown, Unknown
Defense Date:4 October 1990
Record Number:CaltechTHESIS:04112011-134618813
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6296
Deposited By: Tony Diaz
Deposited On:12 Apr 2011 23:07
Last Modified:16 Apr 2021 23:27

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