The Classification of the Finite Simple Groups

Authors

Daniel Gorenstein, Richard Lyons, Ronald Solomon

Type: Book

Publication Date: 1994-11-18

Citations: 707

DOI: https://doi.org/10.1090/surv/040.1

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Abstract

General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.

Locations

Mathematical surveys
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The Classification of the Finite Simple Groups

1980-06-01

Michael Aschbacher

The Classification of Finite Simple Groups

1983-01-01

Daniel Gorenstein

The Classification of Finite Simple Groups

2011-03-09

Michael Aschbacher , Richard K. Lyons , Stephen M. Smith +1 more

The classification of the Finite Simple Groups: an overview

2004-01-01

Javier Otal Cinca

The Classification of Finite Simple Groups: A Progress Report

2018-06-01

Ronald Solomon

An outline of the classification of finite simple groups

1981-01-01

Daniel Gorenstein

The Classification of the Finite Simple Groups, Number 2

2002-11-15

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 4

1999-03-16

Daniel Gorenstein , Richard Lyons , Ronald Solomon

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The Classification of the Finite Simple Groups, Number 5

2002-09-10

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 8

2018-12-12

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 6

2004-12-07

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 9

2021-02-22

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

The Classification of the Finite Simple Groups, Number 10

2023-10-05

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

The Classification of the Finite Simple Groups, Number 7

2018-02-15

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The finite simple groups and their classification

2021-01-01

Michael Aschbacher

An introduction to the classification of finite simple groups

1979-01-01

Michael Aschbacher

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V.7 The Classification of Finite Simple Groups

2010-12-31

Martin W. Liebeck

Book Review: The classification of the finite simple groups

1997-01-01

Michael Aschbacher

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Book Review: The classification of the finite simple groups

1995-10-01

Michael Aschbacher

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A brief history of the classification of the finite simple groups

2001-03-27

Ronald Solomon

We present some highlights of the 110-year project to classify the finite simple groups. We present some highlights of the 110-year project to classify the finite simple groups.

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Sylow d-tori of classical groups and the McKay conjecture, II

2010-02-26

Britta Späth

Characterizations of the solvable radical

2009-12-02

Paul Flavell , Simon D. Guest , Robert M. Guralnick

We prove that there exists a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the property: if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> … We prove that there exists a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the property: if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a conjugacy class of a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generate a solvable subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 4"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 10"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=10</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By lengthening one of our arguments slightly, we obtain a value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 7"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=7</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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From Thompson to Baer–Suzuki: A sharp characterization of the solvable radical

2010-03-12

Nikolai Gordeev , Fritz Grunewald , Boris Kunyavskiı̆ +1 more

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A description of Baer–Suzuki type of the solvable radical of a finite group

2008-07-23

Nikolai Gordeev , Fritz Grunewald , Boris Kunyavskiı̆ +1 more

A classification of finite antiflag-transitive generalized quadrangles

2016-06-01

John Bamberg , Cai Heng Li , Eric Swartz

A generalized quadrangle is a point-line incidence geometry <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that: … A generalized quadrangle is a point-line incidence geometry <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that: (i) any two points lie on at most one line, and (ii) given a line <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not incident with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a unique point of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> collinear with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1–57; Invent. Math. 24 (1974), 191–239], and we study a larger class of generalized quadrangles: the <italic>antiflag-transitive</italic> quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma script l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(P, \ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we say that the generalized quadrangle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is antiflag-transitive, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either a classical generalized quadrangle or is the unique generalized quadrangle of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 3 comma 5 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(3,5)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or its dual. Our approach uses the theory of locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291–317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187–233] on “large” subgroups of simple groups of Lie type to fully classify them.

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Finite groups with odd Sylow normalizers

2016-03-30

Robert M. Guralnick , Gabriel Navarro , Pham Huu Tiep

We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight … We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups at these primes.

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Irreducible restrictions of representations of symmetric and alternating groups in small characteristics

2020-05-13

Alexander Kleshchev , Lucia Morotti , Pham Huu Tiep

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A classification of finite locally 2-transitive generalized quadrangles

2020-07-22

John Bamberg , Cai Heng Li , Eric Swartz

Ostrom and Wagner (1959) proved that if the automorphism group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite projective plane <inline-formula content-type="math/mathml"> … Ostrom and Wagner (1959) proved that if the automorphism group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite projective plane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transitively on the points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to the Desarguesian projective plane and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper P normal upper Gamma normal upper L left-parenthesis 3 comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {P} \Gamma \mathrm {L}(3,q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (for some prime-power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). In the more general case of a finite rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible spherical building, also known as a <italic>generalized polygon</italic>, the theorem of Fong and Seitz (1973) gave a classification of the <italic>Moufang</italic> examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.

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Dan Ciubotaru , Xuhua He

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Sylow subgroups, exponents, and character values

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Gabriel Navarro , Pham Huu Tiep

If<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>is a finite group,<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>is a prime, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>is a Sylow<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-subgroup of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … If<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>is a finite group,<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>is a prime, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>is a Sylow<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-subgroup of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>, we study how the exponent of the abelian group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P slash upper P prime"><mml:semantics><mml:mrow><mml:mi>P</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">P/P’</mml:annotation></mml:semantics></mml:math></inline-formula>is affected and how it affects the values of the complex characters of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>. This is related to Brauer’s Problem<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="12"><mml:semantics><mml:mn>12</mml:mn><mml:annotation encoding="application/x-tex">12</mml:annotation></mml:semantics></mml:math></inline-formula>. Exactly how this is done is one of the last unsolved consequences of the McKay–Galois conjecture.

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2020-02-20

Meinolf Geck , Gunter Malle

After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using … After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.

Index

2020-02-20

Meinolf Geck , Gunter Malle

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Unipotent Characters

2020-02-20

Meinolf Geck , Gunter Malle

After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of … After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.

Arc-transitive digraphs of given out-valency and with blocks of given size

2019-01-11

Luke Morgan , Primož Potočnik , Gabriel Verret

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Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020-08-06

А. С. Кондратьев , Н. В. Маслова , D. O. Revin

Abstract A subgroup H of a group G is said to be pronormal in G if H and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mi>g</m:mi> </m:msup> </m:math> {H^{g}} are conjugate in <m:math … Abstract A subgroup H of a group G is said to be pronormal in G if H and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mi>g</m:mi> </m:msup> </m:math> {H^{g}} are conjugate in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>H</m:mi> <m:mi>g</m:mi> </m:msup> <m:mo stretchy="false">〉</m:mo> </m:mrow> </m:math> {\langle H,H^{g}\rangle} for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> {g\in G} . In this paper, we determine the finite simple groups of type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>E</m:mi> <m:mn>6</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {E_{6}(q)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mmultiscripts> <m:mi>E</m:mi> <m:mn>6</m:mn> <m:none /> <m:mprescripts /> <m:none /> <m:mn>2</m:mn> </m:mmultiscripts> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

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Rank one isolated p-minimal subgroups in finite groups

2020-09-08

Ulrich Meierfrankenfeld , Christopher Parker , Peter Rowley

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Extensions of unipotent characters and the inductive McKay condition

2008-07-27

Gunter Malle

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Characters of odd degree

2016-09-16

Gunter Malle , Britta Späth

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verication of the inductive McKay condition for groups of Lie type and … We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verication of the inductive McKay condition for groups of Lie type and primes ‘ such that a Sylow ‘-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.

EXTENSIONS OF CHARACTERS IN TYPE D AND THE INDUCTIVE MCKAY CONDITION, I

2023-09-08

Britta Späth

Abstract This is a contribution to the study of $\mathrm {Irr}(G)$ as an $\mathrm {Aut}(G)$ -set for G a finite quasisimple group. Focusing on the last open case of groups … Abstract This is a contribution to the study of $\mathrm {Irr}(G)$ as an $\mathrm {Aut}(G)$ -set for G a finite quasisimple group. Focusing on the last open case of groups of Lie type $\mathrm {D}$ and $^2\mathrm {D}$ , a crucial property is the so-called $A'(\infty )$ condition expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in $\mathrm {Irr}(G)$ . This is part of the stronger $A(\infty )$ condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition $A(\infty )$ for groups of type $\mathrm {D}$ would still satisfy $A'(\infty )$ . This will be used in a second paper to fully establish $A(\infty )$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of $G=\mathrm {D}_{ l,\mathrm {sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.

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A generalization of the Brauer–Fowler theorem

2024-03-14

Saveliy V. Skresanov

Abstract The famous Brauer–Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the … Abstract The famous Brauer–Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most 𝑛 involutions, then the group is finite and its order is bounded in terms of 𝑛 only. This answers a question of Strunkov from the Kourovka notebook.

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Support varieties for modules over Chevalley groups and classical Lie algebras

2007-11-09

Jon Carlson , Zongzhu Lin , Daniel K. Nakano

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive algebraic group over an algebraically closed field of characteristic <inline-formula content-type="math/mathml"> <mml:math … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive algebraic group over an algebraically closed field of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 1"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the first Frobenius kernel, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis double-struck upper F Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G({\mathbb F}_{p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the corresponding finite Chevalley group. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a rational <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module. In this paper we relate the support variety of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over the first Frobenius kernel with the support variety of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over the group algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k upper G left-parenthesis double-struck upper F Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">kG({\mathbb F}_{p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.

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On simple endotrivial modules

2011-03-22

Geoffrey R. Robinson

We show that when G is a finite group which contains an elementary Abelian subgroup of order p^2 and k is an algebraically closed field of characteristic p, then the … We show that when G is a finite group which contains an elementary Abelian subgroup of order p^2 and k is an algebraically closed field of characteristic p, then the study of simple endotrivial kG-modules which are not monomial may be reduced to the case when G is quasi-simple.

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Families of explicitly isogenous Jacobians of variable-separated curves

2011-08-01

Benjamin Smith

Abstract We construct six infinite series of families of pairs of curves ( X , Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny … Abstract We construct six infinite series of families of pairs of curves ( X , Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3 or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou-Noguès and Couveignes’ explicit classification of pairs ( f , g ) of polynomials such that f ( x 1 )− g ( x 2 ) is reducible. Supplementary materials are available with this article.

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The Inductive Blockwise Alperin Weight Condition for the Chevalley Groups 𝐹₄(𝑞)

2024-12-10

Jianbei An , Gerhard Hiß , Frank Lübeck

We verify the inductive blockwise Alperin weight condition in odd characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the finite exceptional Chevalley groups … We verify the inductive blockwise Alperin weight condition in odd characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the finite exceptional Chevalley groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 4 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F_4(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not divisible by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

The Character Theory of Finite Groups of Lie Type

2020-02-20

Meinolf Geck , Gunter Malle

Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich … Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich and vast area of mathematics. It incorporates tools and methods from algebraic geometry, topology, combinatorics and computer algebra, and has since evolved substantially. With this book, the authors meet the need for a contemporary treatment, complementing in core areas the well-established books of Carter and Digne–Michel. Focusing on applications in finite group theory, the authors gather previously scattered results and allow the reader to get to grips with the large body of literature available on the subject, covering topics such as regular embeddings, the Jordan decomposition of characters, d-Harish–Chandra theory and Lusztig induction for unipotent characters. Requiring only a modest background in algebraic geometry, this useful reference is suitable for beginning graduate students as well as researchers.

Nonabelian composition factors of a finite group whose all maximal subgroups are hall

2012-09-01

Н. В. Маслова

Morita equivalences and the inductive blockwise Alperin weight condition for type $\mathsf A$

2021-01-01

Zhicheng Feng , Zhenye Li , Jiping Zhang

As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Sp\"ath for simple groups … As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Sp\"ath for simple groups of Lie type $\mathsf A$, split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnaf\'e, Dat and Rouquier.

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Abelian quotients and orbit sizes of finite groups

2014-01-01

Thomas Michael Keller , Yong Yang

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ … Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ has a regular orbit on $V$. In this paper we show that $G$ has an orbit of size at least $|G/G'|$ on $V$. This generalizes earlier work of the authors, where the same bound was proved under the additional hypothesis that $G$ is solvable. For completely reducible modules it also strengthens the 1989 result $|G/G'|

Spectra of Automorphic Extensions of Finite Simple Exceptional Groups of Lie Type

2016-11-01

M. A. Zvezdina

Generation of fusion systems of characteristic 2-type

2009-12-10

Michael Aschbacher

A product decomposition for the classical quasisimple groups

2007-01-26

Nikolay Nikolov

We prove that every quasisimple group of classical type is a product of boundedly many conjugates of a quasisimple subgroup of type An. We prove that every quasisimple group of classical type is a product of boundedly many conjugates of a quasisimple subgroup of type An.

On a relation between the Sylow and Baer-Suzuki theorems

2011-09-01

D. O. Revin

Element orders and Sylow structure of finite groups

2005-08-13

Gunter Malle , Alexander Moretó , Gabriel Navarro

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On Integers that are Covering Numbers of Groups

2019-07-12

Martino Garonzi , Luise-Charlotte Kappe , Eric Swartz

Martino Garonzia , Luise-Charlotte Kappeb & Eric Swartzc* a Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazilb Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, … Martino Garonzia , Luise-Charlotte Kappeb & Eric Swartzc* a Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazilb Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY, USAc Department of Mathematics, College of William & Mary, Williamsburg, VA, USA

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Perfect commuting graphs

2016-06-07

John R. Britnell , Nick Gill

Abstract We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect. Abstract We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.

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Final group topologies, Phan systems and Pontryagin duality

2006-03-22

Helge Glöckner , Ralf Gramlich , Tobias Hartnick

We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_\omega-space, or … We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_\omega-space, or locally k_\omega. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of k_\omega-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally k_\omega topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally k_\omega abelian groups.

Locally finite groups of finite centraliser dimension

2018-04-21

Alexandre Borovik , Ulla Karhumäki

Groups of Finite Morley Rank with a Pseudoreflection Action

2011-12-16

Ayşe Berkman , Alexandre Borovik

In this work, we give two characterisations of the general linear group as a group $G$ of finite Morley rank acting on an abelian connected group $V$ of finite Morley … In this work, we give two characterisations of the general linear group as a group $G$ of finite Morley rank acting on an abelian connected group $V$ of finite Morley rank definably, faithfully and irreducibly. To be more precise, we prove that if the pseudoreflection rank of $G$ is equal to the Morley rank of $V$, then $V$ has a vector space structure over an algebraically closed field, $G\cong GL(V)$ and the action is the natural action. The same result holds also under the assumption of Prufer 2-rank of $G$ being equal to the Morley rank of $V$.

Character theory of finite groups

1999-07-01

I. M. Isaacs

Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur … Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.

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Finite Group Theory

2000-06-26

Michael Aschbacher

During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many … During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.

Representation Theory of Finite Groups

1965-01-01

Walter Feit

SIMPLE GROUPS OF LIE TYPE

1975-03-01

I. G. Macdonald

By W. Carter Roger: pp. viii, 331. £7.50. John Wiley & Sons, December 1972.) By W. Carter Roger: pp. viii, 331. £7.50. John Wiley & Sons, December 1972.)

Homotopy and homology of p-subgroup complexes

1994-01-01

Kaustuv Mukul Das

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis we analyzed the simple connectivity of the Quillen … NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis we analyzed the simple connectivity of the Quillen complex at [...] for the classical groups of Lie type. In light of the Solomon-Tits theorem, we focused on the case where [...] is not the characteristic prime. Given (p,q) = 1. let dp(q) be the order of [...] in [...]. In this thesis we proved the following result: Main Theorem. When (p,q) = 1 we have the following results about the simple connectivity of the Quillen complex at p, Ap(G), for the classical groups of Lie type: 1. If G = GLn(q), dp(q) > 2 and mp(G) > 2, then Ap[...](G) is simply connected. 2. If G = [...], then: (a) Ap(G) is Cohen-Macaulay of dimension n - 1 if dp(q) = 1. (b) If nip(G) > 2 and dp(q) is odd, then Ap(G) is simply connected. 3. If G = [...], then: (a) Ap(G) is Cohen-IVlacaulay of dimension n - 1 if [...] and dp(q) = 1. (b) If mp(G) > 2 and dp(q) is odd, then ,Ap(G) is simply connected. (c) If n [...] 3, q [...] 5 is odd, and dp(q) = 2, then Ap(G)(> Z) is simply connected, where Z is the central subgroup of G of order p. In the course of analyzing the [...]-subgroup complexes we developed new tools for studying relations between various simplicial complexes and generated results about the join of complexes and the [...]-subgroup complexes of products of groups. For example we proved: Theorem A. Let [...] be a map of posets satisfying: (1) [...] is strict; that is,[...] (2)[...] (3) [...]connected for all [...] with [...]. Then Y n-connected implies X is n-connected. Theorem A provides us with a tool for studying [...] in terms of [...]. For example, we used this method to prove: Theorem 8.6. Let G = [...] where [...] is solvable and S is a p-group of symplectic type. Then [...]spherical. In this thesis we also generated a library of results about geometric complexes which do not arise as [...]-subgroup complexes. This library includes, but is not restricted to, the following: (l.) the poset of proper nondegenerate subspaces of a 2[...]-dimensional symplectic space -ordered by inclusion - is Cohen-Macaulay of dimension n-2. (2) If q is an odd prime power anal n [...] (with n [...] 5 if q = 3), then the poset of proper nondegenerate subspaces of an n-dimensional unitary space over Fq2 is simply connected.

Chevalley groups as standard subgroups, I

1979-03-01

Gary M. Seitz

IntroductionLet G be a finite group and A a quasisimple subgroup of G. Then A is called a standard subgroup if K Co(A) is tightly embedded (i.e.IKI is even, but … IntroductionLet G be a finite group and A a quasisimple subgroup of G. Then A is called a standard subgroup if K Co(A) is tightly embedded (i.e.IKI is even, but IKf')KgI is odd for gqN6(K)), No(A)=No(K), and [A, Ag] 1 for all g G.The importance of such subgroups is evident from the work of Aschbacher (see Theorem 1 of [1]).The recent approach to the classification of all finite simple groups requires the determination of those groups, G, having a standard subgroup, A, such that A/Z(A)= A is one of the currently known simple groups.This paper and its sequels are concerned with the case of a group of Lie type defined over a field of characteristic 2. Our results aim at finding the possibilities for G when A has Lie rank at least 3, although we will not treat the cases A -Sp(6, 2), U6(2), or 0+/-(8, 2)'.Our proofs will be inductive so we require information about the rank 1 and rank 2 configurations as well as information about the four cases above.The necessary results, not covered to date, are assembled in the following hypothesis.Hypothesis(*).Let P be quasisimple with [Z(P)I odd and P/Z(P)- Sp(6, 2), U6(2), or 0+(8, 2)'.If P is a standard subgroup of a group X with O(X)= 1 and Cx(P) having cyclic Sylow 2-subgroups, then one of the following occurs"E(X)--Px P. E(X) is a group of Lie type defined over a field of characteristic 2. P 0+(8, 2)' and E(X) M( 22).For a group, X, we set X X/Z(X).Our main result is as follows.MAIN THEOREM.Assume that Hypothesis (*) holds and that the B(G)- conjecture holds.Let A be a quasisimple group with IZ(A)I odd and fi a finite group of Lie type defined over a field o[ characteristic 2 and having Lie rank at least 3. Suppose that A is a standard subgroup of G and that C (A) has

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On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups

1970-01-01

Anne R. MacWilliams

We prove that in a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group with no normal Abelian subgroup of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-over-equals … We prove that in a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group with no normal Abelian subgroup of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-over-equals 3"> <mml:semantics> <mml:mrow> <mml:mo>≧</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\geqq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, every subgroup can be generated by four elements. This result is then used to determine which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with no normal Abelian subgroup of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-over-equals 3"> <mml:semantics> <mml:mrow> <mml:mo>≧</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\geqq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can occur as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{S_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>’s of finite simple groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, under certain assumptions on the embedding of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Finite groups with standard components of Lie type over fields of characteristic two

1983-02-01

Robert H. Gilman , Robert L. Griess

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The orders of the classical simple groups

1955-11-01

Emil Artin

Near solvable signalizer functors on finite groups

1982-09-01

Patrick Paschal McBride

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Solvable signalizer functors on finite groups

1972-04-01

David M. Goldschmidt

On a class of finite simple groups of Ree

1966-09-01

Zvonimir Janko , John G. Thompson

The characterization of finite groups with dihedral Sylow 2-subgroups. I

1965-03-01

Daniel Gorenstein , John H. Walter

On Solvable Signalizer Functors in Finite Groups

1976-07-01

George Glauberman

On the Sims Conjecture and Distance Transitive Graphs

1983-09-01

Peter J‎. Cameron , Cheryl E. Praeger , Jan Saxl +1 more

The local structure of finite groups of characteristic 2 type

1983-01-01

Daniel Gorenstein , Richard Lyons

Part I: Properties of $K$-groups and Preliminary Lemmas: Introduction Decorations of the known simple groups Local subgroups of the known simple groups Balance and signalizers Generational properties of $K$-groups Factorizations … Part I: Properties of $K$-groups and Preliminary Lemmas: Introduction Decorations of the known simple groups Local subgroups of the known simple groups Balance and signalizers Generational properties of $K$-groups Factorizations Miscellaneous general results and lemmas about $K$-groups Appendix by N. Burgoyne Part II: The Trichotomy Theorem: Odd standard form Signalizer functors and weak proper $2$-generated $p$-cores Almost strongly $p$-embedded maximal $2$-local subgroups References.

On transitive permutation groups in which a 2-central involution fixes a unique point

1979-01-01

Fredrick Smith

The Characterization of Finite Groups with Abelian Sylow 2-Subgroups

1969-05-01

John H. Walter

A new finite simple group with abelian Sylow 2-subgroups and its characterization

1966-03-01

Zvonimir Janko

On Janko's Simple Group of Order 50,232,960

1969-03-01

Graham Higman , John McKay

A presentation of the symplectic and orthogonal groups

1979-10-01

Larry Finkelstein , Ronald Solomon

Maximal 2-components in finite groups

1976-01-01

Ronald Solomon

Generators and relations for classical groups

1974-12-01

W. J. Wong

Pushing-up in finite groups

1979-03-01

Richard Niles

Finite groups with a split BN-pair of rank 1. I

1972-03-01

Christoph Hering , William M. Kantor , Gary M. Seitz

Modular representations of split 𝐵𝑁 pairs

1969-01-01

Forrest Richen

Introduction.In 1964 C. W. Curtis classified the absolutely irreducible modular representations of a large class of groups, the so called finite groups of Lie type.This was done by finding in … Introduction.In 1964 C. W. Curtis classified the absolutely irreducible modular representations of a large class of groups, the so called finite groups of Lie type.This was done by finding in each irreducible module an element, called a weight element, which is an eigenvector for certain elements in the modular group algebra; proving that two irreducible modules are isomorphic if and only if the corresponding eigenvalues (the collection of such being called a weight) are equal ; and finally by determining which weights are associated with irreducible modules, i.e. which weights actually occur.This paper is a continuation of Curtis' work.We construct the absolutely irreducible modular representations of a finite group of Lie type by finding weight elements in the modular group algebra which generate a full set of nonisomorphic minimal left ideals.In some work of Steinberg and Curtis (see [14] and [6]) these representations were constructed for the covering groups of the Chevalley groups using the representations of the associated modular Lie algebras.The discussion given here avoids the Lie algebras altogether.Instead we construct each irreducible submodule of certain induced modules by finding the required weight elements.We prove at the same time that these induced modules have multiplicity free socles.Some remarks are made about the related questions of degrees and block structure of these representations.The finite groups of Lie type whose representations were classified by Curtis in [7] were defined by a large number of axioms.These apparently consisted of basic properties possessed by all known examples of such groups, the Chevalley groups and variations thereon as defined by Steinberg [13], for example.A simplified axiom scheme is used here, the heart of which is the axioms for groups with BN pairs (Tits [16]).The groups discussed in [7] satisfy the simplified axioms.The disadvantage of introducing new axioms is that the classification described in the first paragraph of this introduction must be redone.The proofs in this reworking are always similar to Curtis' but rarely identical, for the new axioms are enough weaker that some of the properties Curtis used do not follow from them.The advantages of reworking the classification theorem are twofold.First, the theorem is proved in greater generality.Secondly, the commutator relation is never assumed.It is usually replaced by arguments involving lengths of words in the Weyl group.The effect is to clarify the role of the root system in the classification theorem.

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Components of finite groups

1976-01-01

Robert H. Gilman

Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups

1990-01-01

Wolfgang Kimmerle , Richard Lyons , Robert Sandling +1 more

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's … The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups.

La Géométrie des Groupes Classiques

1963-01-01

Jean Dieudonné

On a theorem of Borel and Tits for finite Chevalley groups

1976-12-01

N. Burgoyne , C. Williamson

On groups generated by three-dimensional special unitary groups II

1977-03-01

Kok-Wee Phan

We shall determine in this paper groups of types D n , E 6 , E 7 and E 8 generated by SU (3, q)'s, q odd, q > 3. … We shall determine in this paper groups of types D n , E 6 , E 7 and E 8 generated by SU (3, q)'s, q odd, q > 3. These groups are defined in Phan (1975). [We shall refer to this paper as I]. Acquaintance with the results of I is assumed. The identification of groups of type D 4 is similar to that of SU (n, q). We actually construct an isomorphism from the universal group of type D 4 onto Spin + (8, q). This direct approach does not appear to be feasible for groups of type D n with n ≧ 5. Fortunately Wong's recent result (1974) is applicable here. But his theorem requires that the characteristic of the field be odd; hence unlike the unitary case, we assume that q is odd and q 3. Using Wong's theorem, we proceed to show by induction that groups of type D n are homomorphic images of Spin + (2n, q) or Spin − (2n, q) according as n is even or n is odd.

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Schur multipliers of finite simple groups of Lie type

1973-01-01

Robert L. Griess

This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let <italic>p</italic> denote the characteristic of the finite field over which such a group is defined. … This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let <italic>p</italic> denote the characteristic of the finite field over which such a group is defined. We determine the <italic>p</italic>-part of the multiplier of the Chevalley groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 2 left-parenthesis 4 right-parenthesis comma upper G 2 left-parenthesis 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_2}(4),{G_2}(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 4 left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_4}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the Steinberg variations; the Ree groups of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 4"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Tits simple group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="squared upper F 4 left-parenthesis 2 right-parenthesis prime"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi /> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">^2{F_4}(2)’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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On Ree’s series of simple groups

1966-01-01

Harold N. Ward

Introduction.The purpose of this paper is to obtain the character tables of the finite simple groups of Ree related to the Lie algebra G2 (presented in [16], [17]) from certain … Introduction.The purpose of this paper is to obtain the character tables of the finite simple groups of Ree related to the Lie algebra G2 (presented in [16], [17]) from certain basic properties of these groups.In the process we shall derive a number of additional properties of the Ree groups.We incorporate the basic properties as conditions in the following definition:Definition.A finite group G will be said to be of Ree type if it satisfies the following five conditions: I.The 2-Sylow subgroups of G are elementary Abelian of order 8. II.G has no normal subgroup of index 2. III.For some element J of order 2 (an "involution") in G, the centralizer CG(J) of J in G is the direct product of <J> and L where L is isomorphic to the linear fractional group LF(2,q).Condition I implies that q = 4 + e (mod 8) where e = + 1.IV.If <R> denotes a cyclic subgroup of order (q + e)/2 in L, then the normalizer NC((R0}) of any subgroup <R0) ¥= <1> of <R> is contained in cG(J).V. Let J' be an involution of L and 5 an element of L of order (q -e)\\ which centralizes J'.Then an element of G of order 3 which normalizes <J, J'} does not centralize S.We call q the characteristic of G.The verification of these conditions is straightforward from the description of Ree's groups in [17].The existence of elements R, J', and S of IV and V is a consequence of the known structure of LF(2,q) which will be summarized in paragraph 1-1.Moreover, there is an involution J" of L commuting with J' for which J"SJ" = S~ , the centralizer of J' in Lis <J',J",5>.Thus the centralizer CG((J,J'}) is (J,J',J",Sy.Also, there is an element of order 3 in L normalizing <J',J"> but not centralizing it.

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Finite groups whose 2-subgroups are generated by at most 4 elements

1974-01-01

Daniel Gorenstein , Koichiro Harada

Automorphisms of unitary block designs

1972-03-01

Michael E. O'Nak

Sylow intersections and fusion

1967-06-01

J. L. Alperin

Large extraspecial subgroups of widths 4 and 6

1979-06-01

Stephen D. Smith

Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen.

1911-01-01

Julia Schur

Article Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. was published on January 1, 1911 in the journal Journal für die reine und angewandte Mathematik … Article Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. was published on January 1, 1911 in the journal Journal für die reine und angewandte Mathematik (volume 1911, issue 139).

Über den größtenp′-Normalteiler inp-auflösbaren Gruppen

1967-01-01

Helmut Bender

The Embedding of 2-Locals in Finite Groups of Characteristic 2-Type, I

1981-09-01

Michael Aschbacher , Daniel Gorenstein , Richard Lyons