On the uniform domination number of a finite simple group

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Let G be a finite simple group. By a theorem of Guralnick and Kantor, G contains a conjugacy class C such that for each non-identity element x ∈ G, there exists y ∈ C with G = hx, yi. Building on this deep result, we introduce a new invariant
γu(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 6= x ∈ G, there exists s ∈ S with G = hx, si. (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 γu(G) 6 |C| for some conjugacy class C of G, and the aim of this paper is to determine close to best possible bounds on γu(G) for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups G with γu(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.
Original languageEnglish
Pages (from-to)545-583
Number of pages39
JournalTransactions of the American Mathematical Society
Issue number1
Early online date2 Nov 2018
Publication statusPublished - 2019


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