Upper Bounds for the Number of Conjugacy Classes of a Finite Group

Liebeck, Martin W.; Pyber, László

doi:10.1006/jabr.1997.7158

Cited by 52 publications

(60 citation statements)

References 29 publications

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“…In the last two cases the extra refinement in the last sentence of Lemma 5.6 gives the conclusion. And in the first, we have ν V (λ i ) (x) ≥ 2 for any non-identity semisimple x ∈ D − 4 (q) (where i = 1, 3 or 4); moreover, if ν V 1 (x) = 2 then ν V i (x) > 2 for i = 3 or 4, and this yields (23).…”

Section: Proof Of Proposition 55mentioning

confidence: 92%

Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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Let Γ be a Fuchsian group of genus at least 2 (at least 3 if Γ is non-oriented). We study the spaces of homomorphisms from Γ to finite simple groups G, and derive a number of applications concerning random generation and representation varieties.Precise asymptotic estimates for |Hom(Γ, G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G| μ(Γ)+1+o(1) , where μ(Γ) is the measure of Γ. We then prove that a randomly chosen homomorphism from Γ to G is surjective with probability tending to 1 as |G| → ∞. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties Hom(Γ,Ḡ), whereḠ is GL n (K) or a simple algebraic group over K, an algebraically closed field of arbitrary characteristic.A key ingredient of our approach is character theory, involving the study of the 'zeta function' ζ G (s) = χ(1) −s , where the sum is over all irreducible complex characters χ of G.

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Section: Proof Of Proposition 55mentioning

confidence: 92%

Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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“…Observe that for χ ∈ Irr(H) nonlinear, |χ(x)| < |C H (x)| 1/2 < q (1−ε/2)n ; moreover, χ(1) ≥ (q n − 1)/2 by [19], and |Irr(H)| < q. (6q) n by [22,Theorem 1]. Hence…”

Section: Lemma 42mentioning

confidence: 98%

“…Hence we may assume t ≥ q r . On the other hand, by [22,Theorem 1], the number of classes k(G) in G is at most q 3r . Hence we have N (t) ≤ t 3 .…”

Section: Reduction To Conjugacy Classesmentioning

confidence: 99%

“…It follows that if x ∈ G is randomly chosen, then with probability at least 1−δ, we have |x G | ≥ δ|G|/k(G). Now for G = A n we have k(G) ≤ P (n) = |G| o(1) (where P (n) is the partition function), and by [22,Theorem 1], for G of Lie type we have k(G) ≤ |G| 1/3+o (1) . Hence |G|/k(G) ≥ |G| 2/3−o (1) , and the conclusion follows by choosing (say) δ = |G| −1/6 .…”

Section: It Follows Thatmentioning

confidence: 99%

“…The main idea in this case is to use a certain trick of merging short Jordan blocks to larger ones, until an element with a very small centralizer is produced. At this stage we apply character ratios with estimates on the number of conjugacy classes [22] to finish the proof. The applications are discussed in Section 8, where Corollary 1.5 and Theorems 1.6, 1.12 and 1.13 are proved.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Diameters of Finite Simple Groups: Sharp Bounds and Applications

Liebeck¹,

Shalev²

2001

The Annals of Mathematics

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115

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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The k(GV) Conjecture for Modules in Characteristic 31

Gluck

Magaard

2002

Journal of Algebra

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Let G be a finite group and let V be a faithful, irreducible, finite G-module, with G V = 1. The k GV problem is to show that k GV , the number of conjugacy classes of the semidirect product GV , is at most V . The k GV problem is a special case of Brauer's famous conjecture on the maximum number of ordinary irreducible characters in a p-block of a finite group.The Robinson-Thompson criterion [RT, Theorem 2] says that k GV ≤ V provided that there exists v ∈ V such that V C G v has a faithful selfdual summand; for convenience, we call such vectors v RT-vectors. This criterion has led to much progress on the k GV problem in recent years. In particular, the k GV conjecture has now been proved whenever the characteristic p of V is not 3, 5, 7, 11, 13, 19, or 31; see [GM, Ri 1,KP].This last result follows from a thorough analysis of the extraspecial and quasisimple cases of the k GV problem. In the extraspecial case, G has a normal extraspecial-type subgroup which acts absolutely irreducibly on V . In the quasisimple case, F * G = QZ G , where Q is quasisimple and acts absolutely irreducibly on V . If p is not one of the seven primes above, then every extraspecial and quasisimple case module in characteristic p contains an RT -vector. If p is one of the seven primes above, however, there exists an extraspecial or quasisimple case module G V such that V contains no RT -vector for G. We call such extraspecial or quasisimple case modules basic non-RT modules. All basic non-RT modules have been determined [GM, Ri 1, Ri 2, KP]. Each has dimension 2, 3, or 4 over GF p .

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Upper Bounds for the Number of Conjugacy Classes of a Finite Group

Cited by 52 publications

References 29 publications

Fuchsian groups, finite simple groups and representation varieties

Fuchsian groups, finite simple groups and representation varieties

Diameters of Finite Simple Groups: Sharp Bounds and Applications

The k(GV) Conjecture for Modules in Characteristic 31

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