On the uniform spread of almost simple linear groups

Burness, Timothy C.; Guest, Simon D.

doi:10.1017/s0027763000010680

Cited by 34 publications

(77 citation statements)

References 43 publications

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“…In [24] (also see [44]), Guralnick and Kantor use probabilistic methods to prove that every non-identity element of a finite simple group G belongs to a generating pair (a group with this strong 2-generation property is said to be 3 2 -generated). See [10,16,28,31] for further results in this direction for simple and almost simple groups. We refer the reader to [15] for a recent survey of related topics concerning the generation of simple groups.…”

Section: Introductionmentioning

confidence: 99%

On the uniform domination number of a finite simple group

Burness

Harper

2018

Trans. Amer. Math. Soc.

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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 = x, y . 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 = x ∈ G, there exists s ∈ S with G = x, s . (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) |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.

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Section: Introductionmentioning

confidence: 99%

On the uniform domination number of a finite simple group

Burness

Harper

2018

Trans. Amer. Math. Soc.

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“…One checks that f is well-defined and bijective (this map is sometimes called the Shintani correspondence). One can also show that f has nice fixed point properties for suitable actions of G and H 0 (see [26,Theorem 2.14], for example). The strategy is to choose an element z ∈ H 0 so that the maximal overgroups of z in H 0 are somewhat restricted; hopefully this will allow us to control the maximal subgroups of G containing a representative y ∈ G 0 x of the corresponding G 0 -class in the coset G 0 x.…”

Section: Moreover the Interplay Between Groups And Graphs Suggests Mmentioning

confidence: 99%

Simple groups, fixed point ratios and applications

Burness

2018

Local Representation Theory and Simple Groups

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The study of fixed point ratios is a classical topic in permutation group theory, with a long history stretching back to the origins of the subject in the 19th century. Fixed point ratios arise naturally in many different contexts, finding a wide range of applications. In this survey article we focus on fixed point ratios for simple groups of Lie type, highlighting some of the main results, applications and related problems.

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“…This allows us to define a well‐defined map