Covering numbers for Chevalley groups

Ellers, Erich W.; Gordeev, Nikolai; Herzog, Marcel

doi:10.1007/bf02810691

Cited by 39 publications

(29 citation statements)

References 17 publications

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“…This is actually proved with an explicit constant in [8], [20]. The next application concerns generation properties.…”

Section: Introductionmentioning

confidence: 89%

“…Since the 1970's there has been considerable interest in covering numbers; see the book [1] and the references therein, as well as the more recent papers [21], [8], [20]. The covering number cn(G) of a finite simple group G is defined to be the maximum of cn(C, G) over all nontrivial classes C in G. While it was shown in the 1970s that cn(A n ) = [n/2] (see [1,Chap.…”

Section: Introductionmentioning

confidence: 99%

“…The covering number cn(G) of a finite simple group G is defined to be the maximum of cn(C, G) over all nontrivial classes C in G. While it was shown in the 1970s that cn(A n ) = [n/2] (see [1,Chap. 3]), it is only recently that sharp bounds on covering numbers of simple groups of Lie type have been given (see [8], [20]). Using Corollary 1.8 it is immediate to derive such bounds.…”

Section: Introductionmentioning

confidence: 99%

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Diameters of Finite Simple Groups: Sharp Bounds and Applications

Liebeck¹,

Shalev²

2001

The Annals of Mathematics

113

103

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Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph Γ(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of arbitrary order). We also show that for any word w = w(x 1 , . . . , x d ), there is a constant c = c(w) such that for any simple group G on which w does not vanish, every element of G is a product of c values of w. From this we deduce that every verbal subgroup of a semisimple profinite group is closed. Other applications concern covering numbers, expanders, and random walks on finite simple groups.

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“…This is actually proved with an explicit constant in [8], [20]. The next application concerns generation properties.…”

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Diameters of Finite Simple Groups: Sharp Bounds and Applications

Liebeck¹,

Shalev²

2001

The Annals of Mathematics

113

103

View full text Add to dashboard Cite

show abstract

“…The bound in the proposition is quadratic in the rank; this is not the correct order of magnitude -linear bounds can be found in [10,22]. A general upper bound width(G, g) < C log |G| log |g G | is proved in [28], where C is an absolute constant.…”

Section: Applicationsmentioning

confidence: 99%

Character ratios for finite groups of Lie type, and applications

Liebeck¹

2017

Finite Simple Groups: Thirty Years of The Atlas and Beyond

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“…Let g be a semisimple element of G that has no eigenvalues in K. Then g is a regular element and therefore its conjugacy class C g intersects the big Bruhat cell BẇB (see [EGH,Lemma 4]). But C g ∩ Bẇ B = ∅ because every element of the form b 1ẇ b 2 is conjugate to an element of the forṁ…”

Section: Proofmentioning

confidence: 99%