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 ∈ 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 nonabelian 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.
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 nonabelian 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 language  English 

Pages (fromto)  545583 
Number of pages  39 
Journal  Transactions of the American Mathematical Society 
Volume  372 
Issue number  1 
Early online date  2 Nov 2018 
DOIs  
Publication status  Published  2019 
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Profiles

Dr Tim C Burness
 School of Mathematics  Reader in Pure Mathematics
 Pure Mathematics
 Algebra
Person: Academic , Member
