Arithmetical conditions of orbit sizes of linear groups of odd order
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ARITHMETICAL CONDITIONS OF ORBIT SIZES OF LINEAR GROUPS OF ODD ORDER BY
Yong Yang Department of Mathematics, Texas State University 601 University Drive, San Marcos, TX 78666, USA and Key Laboratory of Group and Graph Theories and Applications Chongqing University of Arts and Sciences, Chongqing 402160, China e-mail: [email protected]
ABSTRACT
We study a conjecture of Gluck and a conjecture of Navarro for groups of odd order. We use some innovative techniques to obtain bounds beyond what is known.
1. Introduction Let G be a finite group that acts faithfully and completely reducibly on a finite vector space V . There are a number of papers devoted to showing that, under certain circumstances, G has a “large” orbit on V . For instance, Espuelas proved in [3] that if |G||V | is odd, then there exists v ∈ V such that |G : CG (v)| ≥ |G|1/2 . It is worth remarking that this is the bound that has often been achieved (see, for instance, [2, 3, 10]). This is due to the fact that these results rely on proving the existence of regular orbits on V ⊕ V . In this paper, we provide a novel approach and are able to improve on Espuelas’ theorem. Theorem 1: Let G be a non-trivial finite group that acts faithfully and completely reducibly on a finite vector space V . If |G||V | is odd, then there exists v ∈ V such that |G : CG (v)| ≥ (1.397) · |G|0.608 . Received October 1, 2018 and in revised form March 24, 2019
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Y. YANG
Isr. J. Math.
If we only assume that |G| is odd, we can also slightly improve the |G|1/2 bound to |G|0.541 . We provide some examples (Remarks 3.4 and 3.5) to show that the bounds we obtained can be attained, but it remains an interesting problem to see if the exponent of |G| can be further increased. As usual, the main motivation to study these problems on orbit sizes comes from problems on character degrees and conjugacy class sizes. For instance, Espuelas’ theorem was used in the same paper [3] to prove Gluck’s conjecture for groups of odd order. We recall that Gluck’s conjecture asserts that if G is a finite solvable group, then |G : F(G)| ≤ b(G)2 . Here F(G) is the Fitting subgroup of G and b(G) is the largest character degree of G. Gluck’s conjecture was also proved for solvable groups with abelian Sylow 2-subgroups by Dolfi and Jabara [2] and for solvable groups with order not divisible by 3 by the author [10]. In this paper, we use Theorem 1 to improve on the bound predicted by Gluck’s conjecture when G has odd order. We show some other applications at the end of the paper, for instance, to a conjecture of Navarro. The paper is organized as follows: In Section 2, we fix some notations and prove some related results about permutation groups acting on the power set. In Section 3, we prove the main results about the orbit structure of linear group actions. In Section 4, we apply those orbit results to study some problems on character degrees and conjugacy class sizes.
2. Notation and lemmas Notation: (1) If V is a finite vector space of dimension n over GF(q), where q is a prime power, we denote by Γ(q n )
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