On a Class of Doubly Transitive Groups

Suzuki, Michio

doi:10.2307/1970423

Cited by 688 publications

(457 citation statements)

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“…The irreducible character degrees and their multiplicities for 2 B 2 (q) and 3 D 4 (q) can be found in [47,5], and for G 2 (q) a summary can be found in [43,Appendix]. Inspection of this data shows that ζ G (s) → 1 for s > s 0 , where…”

Section: Corollary 24mentioning

confidence: 98%

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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: Corollary 24mentioning

confidence: 98%

“…Proof For G of type 2 B 2 , 2 G 2 or 2 F 4 , all the assertions are readily checked using the complete information on conjugacy classes of these groups found in [47,51,44]. So assume from now on that G is not of one of these types.…”

Section: Theorem 43mentioning

confidence: 99%

Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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“…They were first described in [18,19] and [21] respectively, and a good account of their properties can also be found in Chapter XI of [11].…”

Section: Lemma 11mentioning

confidence: 99%

Chirality groups of maps and hypermaps

et al. 2008

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Abstract. Although the phenomenon of chirality appears in many investigations of maps and hypermaps no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants -the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are called totally chiral. Examples of them are constructed by considering appropriate "asymmetric" pairs of generators for some non-abelian simple groups. We also show that every finite abelian group is the chirality group of some hypermap, whereas many non-abelian groups, including symmetric and dihedral groups, cannot arise as chirality groups.

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“…For all these groups the structure of maximal subgroups is known (see [5] (Theorems 2.3 and 2.4), [11] (Theorems A and C), [12] (Theorem 1), [15] (Main Theorem), [19] (Theorem 9)).…”

Section: Proof For Exceptional Lie Type Groupsmentioning

confidence: 99%

Solvability of generalized monomial groups

König¹

2010

Journal of Group Theory

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The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is induced from a character ϕ of U with solvable factor group U/ker(ϕ). Using the classification of finite simple groups, we prove that these groups are also solvable. This means in particular that the mentioned properties do not enable one to deduce a proof of the famous Artin conjecture for any non-solvable group from a possible proof for solvable groups.

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On a Class of Doubly Transitive Groups

Cited by 688 publications

References 0 publications

Fuchsian groups, finite simple groups and representation varieties

Fuchsian groups, finite simple groups and representation varieties

Chirality groups of maps and hypermaps

Solvability of generalized monomial groups

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