On the uniform domination number of a finite simple group
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- by Timothy C. Burness and Scott Harper PDF
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Abstract:
Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each nonidentity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep result, we introduce a new invariant $\gamma _u(G)$, which we call the uniform domination number of $G$. This is the minimal size of a subset $S$ of conjugate elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G = \langle x, s\rangle$. (This invariant is closely related to the total domination number of the generating graph of $G$, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have $\gamma _u(G) \leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper is to determine close to best possible bounds on $\gamma _u(G)$ for each family of simple groups. For example, we will prove that there are infinitely many nonabelian simple groups $G$ with $\gamma _u(G) = 2$. To do this, we develop a probabilistic approach based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.References
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Additional Information
- Timothy C. Burness
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- MR Author ID: 717073
- Email: t.burness@bristol.ac.uk
- Scott Harper
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- MR Author ID: 1229326
- Email: scott.harper@bristol.ac.uk
- Received by editor(s): October 19, 2017
- Received by editor(s) in revised form: February 21, 2018, March 10, 2018, and March 21, 2018
- Published electronically: November 2, 2018
- Additional Notes: The second author thanks the Engineering and Physical Sciences Research Council and the Heilbronn Institute for Mathematical Research for their financial support.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 545-583
- MSC (2010): Primary 20E32, 20F05; Secondary 20E28, 20P05
- DOI: https://doi.org/10.1090/tran/7593
- MathSciNet review: 3968779