Michael Aschbacher

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On the maximal subgroups of the finite classical groups

1984-10-01

Michael Aschbacher

Involutions in Chevalley groups over fields of even order

1976-10-01

Michael Aschbacher , Gary M. Seitz

Let G = G(q) be a Chevalley group defined over a field F q of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut( G … Let G = G(q) be a Chevalley group defined over a field F q of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut( G ) and the centralizers of these involutions. This study was begun in the context of a different problem.

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Maximal subgroups of finite groups

1985-01-01

Michael Aschbacher , Leonard L. Scott

The Classification of the Finite Simple Groups

1980-06-01

Michael Aschbacher

Some applications of the first cohomology group

1984-10-01

Michael Aschbacher , Robert M. Guralnick

A Characterization of Chevalley Groups Over Fields of Odd Order

1977-11-01

Michael Aschbacher

THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian … THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian Sylow 2-subgroups and z is the unique involution in K. Assume for each 2-element k C K that kG n C(z) C N(K) and for each g e C(z) N(K), that [K, Kg]<! O(C(z)). Then F*(G) is a Chevalley group of odd characteristic, M11, M12 or SP6(2). COROLLARY II. Let G be a finite group with F*(G) simple and let K be tightly embedded in G such that K has quaternion Sylow 2-subgroups. Then F*(G) is a Chevalley group of odd characteristic, M11, or M12. COROLLARY III. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a 2-component or solvable 2-component of CQ(z) of 2-rank 1, containing z. Then F*(G) is a Chevalley group of odd characteristic or M,1. Theorem I follows from Theorems 1 through 8, stated in Section 2, which supply more specific information under more general hypotheses. Corollary II follows directly from Theorem I. Corollary III follows from Theorem I and Theorem 3 in [3]. All Chevalley groups with the exception of L2(q) and 2G2(q) satisfy the hypotheses of Theorem I. Terminology and notation are defined in Section 2. The possibility of such a theorem was first suggested by J.G. Thompson in January 1974, during his lectures at the winter meeting of the American Mathematical Society in San Francisco. At the same time Thompson also pointed out the significance of a certain section of the group, which is crucial to the proof. We have taken the liberty of referring to this section as the Thompson group of G; see Section 2 for its definition. The theorem finds its motivation in the study of component-type groups. Some applications to this theory are described in [6]. The remainder of this

Chevalley groups of type G2 as the group of a trilinear form

1987-08-01

Michael Aschbacher

On finite groups of component type

1975-03-01

Michael Aschbacher

In recent years great progress has been made toward the classification of finite simple groups in terms of local subgroups and in particular the centralizers of involutions.If this program is … In recent years great progress has been made toward the classification of finite simple groups in terms of local subgroups and in particular the centralizers of involutions.If this program is to be completed one must show that an arbitrary simple group G possesses an involution for which Ca(t) is isomorphic to a centralizer in a known simple group.This paper concerns itself with that problem for simple groups of component type; that is groups G such that E(C(t)/O(C(t)))for some involution in G.These include most of the Chevalley groups of odd characteristic, most of the alternating groups, and many of the sporadic simple groups.D. Gorenstein has conjectured that in a group of component type, the centralizer of some involu- tion is usually in a "standard form."A proof is supplied here of a portion of that conjecture.To be more precise, define a subgroup K of a finite group G to be tightly embedded in G if K has even order while K c K g has odd order for each g G-N(K).Define a quasisimple subgroup .4 of G to be standard in G if [.4,.4g] for each g G, K Ca(,4) is tightly embedded in G, and N(A) N(K).Let G be a finite simple group of component type in which O,,(C(t)) O(C(t))E(C(t)) for each involution in G. Let A be a "large component."Then it is shown, modulo a certain special case where A has 2-rank l, that A is standard in G in the sense defined above.Other theorems establish properties of tightly embedded subgroups.They show that, under the hypothesis of the last paragraph, the centralizer of each involution centralizing A contains at most one component distinct from A, and that component must have 2-rank if it exists.Further, it can be shown that the 2-rank of the centralizer of A is bounded by a function of A, which seems to be or 2 if A is not of even characteristic.Proofs of the various theorems utilize properties of the Generalized Fitting Subgroup F*(G) of a group G, developed by Gorenstein and Walter.These properties appear in Section 2. Also important to the proof is the classification of groups with dihedral Sylow 2-groups, Alperin's fusion theorem, the recent result on 2-fusion due to Goldschmidt, and Theorem 3.3 in Section 3, which extends Bender's classification of groups with a strongly embedded subgroup.Statements of the major theorems appear in Section l, along with a brief explanation of notation.

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A Characterization of Chevalley Groups Over Fields of Odd Order

1977-09-01

Michael Aschbacher

The generalized Fitting subsystem of a fusion system

2010-12-20

Michael Aschbacher

The notion of a fusion system was first defined and explored by Puig, in the context of modular representation theory. Later, Broto, Levi, and Oliver extended the theory and used … The notion of a fusion system was first defined and explored by Puig, in the context of modular representation theory. Later, Broto, Levi, and Oliver extended the theory and used it as a tool in homotopy theory. The author seeks to build a local theory of fusion systems, analogous to the local theory of finite groups, involving normal subsystems and factor systems. Among other results, he defines the notion of a simple system, the generalized Fitting subsystem of a fusion system, and prove the L-balance theorem of Gorenstein and Walter for fusion systems. He defines a notion of composition series and composition factors and proves a Jordon-Hoelder theorem for fusion systems.|The notion of a fusion system was first defined and explored by Puig, in the context of modular representation theory. Later, Broto, Levi, and Oliver extended the theory and used it as a tool in homotopy theory. The author seeks to build a local theory of fusion systems, analogous to the local theory of finite groups, involving normal subsystems and factor systems. Among other results, he defines the notion of a simple system, the generalized Fitting subsystem of a fusion system, and prove the L-balance theorem of Gorenstein and Walter for fusion systems. He defines a notion of composition series and composition factors and proves a Jordon-Hoelder theorem for fusion systems.

Flag structures on tits geometries

1983-03-01

Michael Aschbacher

On groups with a standard component of known type. II

1981-01-01

Michael Aschbacher , Gary M. Seitz

On finite groups generated by odd transpositions. I

1972-01-01

Michael Aschbacher

Normal subsystems of fusion systems

2008-03-21

Michael Aschbacher

The notion of a fusion system was first defined and explored by Puig in the context of modular representation theory. Later, Broto, Levi, and Oliver significantly extended the theory of … The notion of a fusion system was first defined and explored by Puig in the context of modular representation theory. Later, Broto, Levi, and Oliver significantly extended the theory of fusion systems as a tool in homotopy theory. In this paper we begin a program to establish a local theory of fusion systems similar to the local theory of finite groups. In particular, we define the notion of a normal subsystem of a saturated fusion system, and prove some basic results about normal subsystems and factor systems.

On abelian quotients of primitive groups

1989-01-01

Michael Aschbacher , Robert M. Guralnick

It is shown that 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 primitive permutation group on a set of size <inline-formula content-type="math/mathml"> … It is shown that 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 primitive permutation group on a set of size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then any abelian quotient 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> has order at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a minimal extension and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an abelian extension contained in the normal closure 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>, then the degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is at most the degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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On finite groups of Lie type and odd characteristic

1980-10-01

Michael Aschbacher

Overgroups of Sylow subgroups in sporadic groups

1986-01-01

Michael Aschbacher

Simple connectivity ofp-group complexes

1993-06-01

Michael Aschbacher

On Quillen's Conjecture for the p-Groups Complex

1993-05-01

Michael Aschbacher , Stephen D. Smith

The Classification of Finite Simple Groups

2011-03-09

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

The limitations of nice mutually unbiased bases

2006-07-10

Michael Aschbacher , Andrew M. Childs , Paweł Wocjan

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The nonexistence of rank three permutation groups of degree 3250 and subdegree 57

1971-12-01

Michael Aschbacher

A characterization of the unitary and symplectic groups over finite fields of characteristic at least 5

1973-07-01

Michael Aschbacher

The following characterization is obtained: THEOREM.Let G be a finite group generated by a conjugacy class D of subgroups of prime order p ^ 5, such that for any choice … The following characterization is obtained: THEOREM.Let G be a finite group generated by a conjugacy class D of subgroups of prime order p ^ 5, such that for any choice of distinct A and B in D, the subgroup generated by A and B is isomorphic to Z p x Z p , L 2 (p m ) or SL 2 (p m ), where m depends on A and B. Assume G has no nontrivial solvable normal subgroup.Then G is isomorphic to Sp n (q) or U n (q) for some power q of p.

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Groups generated by a class of elements of order 3

1973-03-01

Michael Aschbacher , Marshall Hall

Finite groups with a proper 2-generated core

1974-01-01

Michael Aschbacher

H. Bender’s classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite … H. Bender’s classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite groups containing subgroups with similar but somewhat weaker embedding properties. The first theorem, classifying the groups of the title, is useful in connection with signalizer functor theory. The second theorem classifies a certain subclass of the class of finite groups possessing a permutation representation in which some involution fixes a unique point.

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Near subgroups of finite groups

1998-04-01

Michael Aschbacher

Some Multilinear Forms with Large Isometry Groups

1988-01-01

Michael Aschbacher

On finite groups generated by odd transpositions, II

1973-09-01

Michael Aschbacher

Tits geometries over gf(2) Defined by groups over gf(3)

1983-01-01

Michael Aschbacher , Stephen D. Smith

Solvable generation of groups and Sylow subgroups of the lower central series

1982-07-01

Michael Aschbacher , Robert M. Guralnick

The 27-dimensional module for E6, IV

1990-05-01

Michael Aschbacher

Thin finite simple groups

1978-09-01

Michael Aschbacher

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Finite geometries of type C 3 with flag-transitive groups

1984-06-01

Michael Aschbacher

On finite groups in which the generalized fitting group of the centralizer of some involution is extraspecial

1977-06-01

Michael Aschbacher

A finite group G is said to be of characteristic 2 type if F*(CG(t)) is a 2-group for each involution in G.It seems probable that in the near future the … A finite group G is said to be of characteristic 2 type if F*(CG(t)) is a 2-group for each involution in G.It seems probable that in the near future the problem of determining the finite simple groups will be reduced to determining the simple groups of characteristic 2 type.The principal model for investigation of the characteristic 2 type groups is Thompson's work on N-groups.There Thompson argues on abelian normal subgroups of 2-locals.As an extreme case, he must consider the situation where, for some maximal 2-local M, abelian normal subgroups of M have order at most 2. Hence Z(M) (z) is of order 2, M CG(z), and F*(M) is an extraspecial 2-group.Since many of the sporadic simple groups possess such centralizers, it seems likely that this will be a trouble- some case in most suitably general classification problems.Thompson's analysis of this situation may be divided into two sections.In Lemma 13.63 he proves that z is weakly closed in F*(M).The remainder of Section 13 is then devoted to eliminating this case The following theorem supplies this latter analysis in general.THEOREM.Let G be a finite group and z an involution in G such that F*(CG(z)) Q is an extraspecial 2-group of width at least 2. Then one of the following hoM: () z z().(2) (z ) F*(G) is isomorphic, to Us(2), the m-dimensional unitary group over GF(2), or the second Conway group Co2.(3) z 02(C(t)) for some involution Q.In particular F*(CG(t)) is not a 2-group, so G is not of characteristic 2 type.(4) z is fused in G to some noncentral involution of Q.The proof depends upon work of B. Fischer and F. Timmesfeld on groups generated by {3, 4}-transpositions.See [6] or [1] for notation and termi- nology.Certain results in Sections 4 and 5, in particular Lemma 5.12, may be of independent interest.Co2 is identified using a result of F. Smith [14].

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Generation of fusion systems of characteristic 2-type

2009-12-10

Michael Aschbacher

On Bol loops of exponent 2

2005-04-10

Michael Aschbacher

GF(2)-Representations of Finite Groups

1982-08-01

Michael Aschbacher

Doubly transitive groups in which the stabilizer of two points is Abelian

1971-05-01

Michael Aschbacher

On conjectures of Guralnick and Thompson

1990-12-01

Michael Aschbacher

Extending Morphisms of Groups and Graphs

1992-03-01

Michael Aschbacher , Yoav Segev

Fusion systems

2016-06-29

Michael Aschbacher , Bob Oliver

This is a survey article on the theory of fusion systems, a relatively new area of mathematics with connections to local finite group theory, algebraic topology, and modular representation theory. … This is a survey article on the theory of fusion systems, a relatively new area of mathematics with connections to local finite group theory, algebraic topology, and modular representation theory. We first describe the general theory and then look separately at these connections.

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Tightly embedded subgroups of finite groups

1976-09-01

Michael Aschbacher

A condition for the existence of a strongly embedded subgroup

1973-05-01

Michael Aschbacher

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a normal set of involutions in a finite group. Form the involutory graph with vertex … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a normal set of involutions in a finite group. Form the involutory graph with vertex set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by joining distinct commuting elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assume the product of any two such elements is in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the graph is disconnected. Then the group generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a strongly embedded subgroup. Two corollaries are proved.

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A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver

2010-03-25

Michael Aschbacher , Andrew Chermak

We extend the notion of a p-local finite group (defined in [BLO03]) to the notion of a p-local group.We define morphisms of p-local groups, obtaining thereby a category, and we … We extend the notion of a p-local finite group (defined in [BLO03]) to the notion of a p-local group.We define morphisms of p-local groups, obtaining thereby a category, and we introduce the notion of a representation of a p-local group via signalizer functors associated with groups.We construct a chain G Dfinite group of the third Conway sporadic group Co 3 , and for n > 0, Ᏻ n is one of the 2-local finite groups constructed by Levi and Oliver in [LO02].We show that the direct limit Ᏻ of G exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain G .The 2-completed classifying space of Ᏻ is shown to be the classifying space B D I.4/ of the exotic 2-compact group of Dwyer and Wilkerson [DW93]. 2-LOCAL FINITE GROUPSThe next lemma states a weak form of the Alperin-Goldschmidt fusion theorem [Gol70], in the language of fusion systems.This result will be of use in the proof of Theorem B. LEMMA 1.9.Let G be a finite group, S 2 Syl p .G/, and denote by ᏺ the set of subgroups N of G having the following properties.

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Corrections to “Involutions in Chevalley groups over fields of even order”

1978-12-01

Michael Aschbacher , Gary M. Seitz

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ … An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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The 27-Dimensional Module for E6 , II

1988-04-01

Michael Aschbacher

A homomorphism theorem for finite graphs

1976-01-01

Michael Aschbacher

A homomorphism theorem for finite graphs is established and several group theoretical applications are derived. The proof is taken from a lemma of B. Fischer in his paper classifying groups … A homomorphism theorem for finite graphs is established and several group theoretical applications are derived. The proof is taken from a lemma of B. Fischer in his paper classifying groups generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transpositions.

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Finite Bruck loops

2005-09-22

Michael Aschbacher , Michael Kinyon , J. D. Phillips

Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> … Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, showing that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is essentially the direct product of a Bruck loop of odd order with a <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>-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all 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>-element Bruck loops are <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>-loops, leaving open the question of whether such obstructions actually exist.

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On Fusion Systems of Component Type

2019-01-01

Michael Aschbacher

On a conjecture of Quillen and a lemma of Robinson

1990-09-01

Michael Aschbacher , Peter B. Kleidman

Intrinsic components in involution centralizers of fusion systems

2025-05-26

Michael Aschbacher

The 2‐fusion system of the Monster

2023-07-19

Michael Aschbacher

Abstract This paper is part of an effort to determine a certain class of simple 2‐fusion systems, and to use that result to simplify the proof of the classification of … Abstract This paper is part of an effort to determine a certain class of simple 2‐fusion systems, and to use that result to simplify the proof of the classification of the finite simple groups. The main theorem proves that the 2‐fusion system of the Monster is the unique simple system with a fully centralized involution whose centralizer is the fusion system of the universal covering group of the Baby Monster.

Fusion systems with 2-small components

2022-09-14

Michael Aschbacher

We show that there are no odd simple 2-fusion systems <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … We show that there are no odd simple 2-fusion systems <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in which the centralizer of some fully centralized involution contains a component <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> that is the 2-fusion system of a simple group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <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 J-maximal or maximal and subintrinsic in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C left-parenthesis script upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {C}(\mathcal {F})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as appropriate, and such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of Lie type over the field of order 2, but not <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p Subscript n Baseline left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <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">Sp_n(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <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:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <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>; or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is one of many sporadic groups; or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P normal upper Omega 8 Superscript plus Baseline left-parenthesis 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:msubsup> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mn>8</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> <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">P\Omega _8^+(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Fusion systems with J-components over 𝐹2𝑒 with 𝑒 > 1

2022-08-20

Michael Aschbacher

Abstract Let 𝐾 be a finite simple group of Lie type over a field of even order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> q>2 . If 𝐾 is … Abstract Let 𝐾 be a finite simple group of Lie type over a field of even order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> q>2 . If 𝐾 is not <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mmultiscripts> <m:mi>F</m:mi> <m:mn>4</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}F_{4}(q) , then we determine the fusion systems ℱ of J-component type with a fully centralized involution 𝑗 such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi mathvariant="script">F</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>j</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C_{\mathcal{F}}(j) has a component realized by 𝐾. The exceptional case is treated in a later paper.

Fusion systems with U3(3) J-components

2021-06-09

Michael Aschbacher

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The finite simple groups and their classification

2021-01-01

Michael Aschbacher

Walter's basic theorem for fusion systems

2020-12-08

Michael Aschbacher

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Walter's theorem for fusion systems

2020-10-23

Michael Aschbacher

Fusion systems with alternating J‐components

2020-05-14

Michael Aschbacher

We essentially determine the saturated 2-fusion systems of J-component type in which the centralizer of some fully centralized involution of maximal 2-rank contains a component that is the 2-fusion system … We essentially determine the saturated 2-fusion systems of J-component type in which the centralizer of some fully centralized involution of maximal 2-rank contains a component that is the 2-fusion system of an alternating group A n for some n ⩾ 8 .

The 2-fusion system of an almost simple group

2019-08-22

Michael Aschbacher

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Finite Simple Groups and Fusion Systems

2019-04-11

Michael Aschbacher

We’ll begin with an introduction to the basic theory of fusion systems. Then we give an overview of the proof of that part of the CFSG devoted to the groups … We’ll begin with an introduction to the basic theory of fusion systems. Then we give an overview of the proof of that part of the CFSG devoted to the groups of component type, after which we discuss how to translate that proof into the category of 2-fusion systems, and indicate some advantages that accrue from that translation. We also describe some other changes to the original proof of the CFSG that are part of the program.

On Fusion Systems of Component Type

2019-01-01

Michael Aschbacher

A characterization of ${}^2\mathrm{E}_6(2)$

2018-12-29

Michael Aschbacher

A characterization of 2 E6( A characterization of 2 E6(

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Combinatorial Cell Complexes

2018-08-15

Michael Aschbacher

We define and discuss a category of combinatorial objects we call combinatorial cell complexes and a functor T from this category to the category of topological spaces with cell structure, … We define and discuss a category of combinatorial objects we call combinatorial cell complexes and a functor T from this category to the category of topological spaces with cell structure, whose image is closely related to the category of CW-complexes.This formalism was developed to study finite group actions on topological spaces.In order to make effective use of our detailed knowledge of the finite simple groups, it seems necessary to make such a translation from a purely topological setting to the language of geometric combinatorics.Our functor T assigns to each combinatorial cell complex X its geometric realization T(X).We show the functor T defines an equivalence of categories between the category of combinatorial cell complexes whose cell boundaries are spheres, and a certain subcategory of CW-complexes we call normal CW-complexes.We often concentrate on a subcategory of combinatorial cell complexes we call restricted combinatorial cell complexes; the restricted CW-complexes are the CW-complexes corresponding to the restricted combinatorial cell complexes under our equivalence of categories.Restricted CW-complexes include regular CW-complexes but also many other classical examples like the torus, the Klein bottle, and the Poincare dodecahedron, which are discussed here as illustrations.We associate to each restricted combinatorial cell complex X, a simplicial complex K(X) and a canonical triangulation of T(X) by K(X).The geometric realization of a general combinatorial cell complex can also be canonically triangulated, but by a more complicated simplicial complex than K (X).However we do not supply a proof of this last fact here.We define cellular homology combinatorially, and show that if X is restricted and the boundary of each cell is homologically-spherical, then

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The subgroup structure of finite groups

2017-01-01

Michael Aschbacher

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Fusion systems

2016-06-29

Michael Aschbacher , Bob Oliver

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Overgroups of root groups in classical groups

2015-12-10

Michael Aschbacher

We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this … We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.

N-groups and fusion systems

2015-11-17

Michael Aschbacher

Classifying Finite Simple Groups and 2-Fusion Systems

2015-01-01

Michael Aschbacher

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Mathematicians and Mathematics

2015-01-01

Michael Aschbacher

Finite groups of Seitz type

2013-10-04

Michael Aschbacher

We show that a useful condition of Seitz on finite groups of Lie type over fields of order $q>4$ is often satisfied when $q$ is $2$ or $3$. We also … We show that a useful condition of Seitz on finite groups of Lie type over fields of order $q>4$ is often satisfied when $q$ is $2$ or $3$. We also observe that various consequences of the Seitz condition, established by Seitz and Cline, Parshall, and Scott when $q>4$, also hold when $q$ is $3$ or $4$.

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Overgroup lattices in finite groups of Lie type containing a parabolic

2013-03-06

Michael Aschbacher

Fusion systems of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-type

2013-01-23

Michael Aschbacher

S3-free 2-fusion systems

2012-12-05

Michael Aschbacher

Abstract We develop a theory of 2-fusion systems of even characteristic, and use that theory to show that all S 3 -free saturated 2-fusion systems are constrained. This supplies a … Abstract We develop a theory of 2-fusion systems of even characteristic, and use that theory to show that all S 3 -free saturated 2-fusion systems are constrained. This supplies a new proof of Glauberman's Theorem on S 4 -free groups and its various corollaries.

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Lower signalizer lattices in alternating and symmetric groups

2012-01-01

Michael Aschbacher

Abstract. We prove that the subgroup lattices of finite alternating and symmetric groups do not contain so-called lower signalizer lattices in the class Abstract. We prove that the subgroup lattices of finite alternating and symmetric groups do not contain so-called lower signalizer lattices in the class

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The local theory of fusion systems

2011-08-25

Michael Aschbacher , Radha Kessar , Bob Oliver

Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension … Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite p-groups.

Fusion and homotopy theory

2011-08-25

Michael Aschbacher , Radha Kessar , Bob Oliver

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.

Introduction

2011-08-25

Michael Aschbacher , Radha Kessar , Bob Oliver

Let G be a finite group, p a prime, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate … Let G be a finite group, p a prime, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate under some element of G. The term “fusion” seems to have been introduced by Brauer in the fifties, but the general notion has been of interest for over a century. For example, in his text The Theory of Groups of Finite Order [Bu] (first published in 1897), Burnside proved that if S is abelian then the normalizer in G of S controls fusion in S. (A subgroup H of G is said to control fusion in S if any pair of tuples of elements of S which are conjugate in G are also conjugate under H.)

𝑒(𝐺)≤2: The classification of quasithin groups

2011-03-09

Michael Aschbacher , Richard Lyons , Stephen D. Smith +1 more

Overview: The classification of groups of Gorenstein-Walter type

2011-03-09

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

Some background material related to simple groups

2011-03-09

Michael Aschbacher , Richard Lyons , Stephen D. Smith +1 more

𝑒(𝐺)≥4: The pretrichotomy and trichotomy theorems

2011-03-09

Michael Aschbacher , Richard Lyons , Stephen D. Smith +1 more

The Classification of Finite Simple Groups

2011-03-09

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

The generalized Fitting subsystem of a fusion system

2010-12-20

Michael Aschbacher

Uniqueness of sporadic groups

2010-09-01

Michael Aschbacher , Yoav Segev

Initial work on the sporadic finite simple groups falls into one or more of the following categories: Initial work on the sporadic finite simple groups falls into one or more of the following categories:

A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver

2010-03-25

Michael Aschbacher , Andrew Chermak

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Generation of fusion systems of characteristic 2-type

2009-12-10

Michael Aschbacher

OVERGROUPS OF PRIMITIVE GROUPS

2009-08-01

Michael Aschbacher

We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. … We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. We then show that certain weak restrictions on the lattice 𝒪G(H) impose strong restrictions on H and its overgroup lattice.

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Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups

2009-07-24

Michael Aschbacher , John Shareshian

Overgroups of primitive groups, II

2009-05-31

Michael Aschbacher

Signalizer lattices in finite groups

2009-05-01

Michael Aschbacher

Let G be a finite group and let H be a subgroup of G. We investigate constraints imposed upon the structure of G by restrictions on the lattice O_G(H) of … Let G be a finite group and let H be a subgroup of G. We investigate constraints imposed upon the structure of G by restrictions on the lattice O_G(H) of overgroups of H in G. Call such a lattice a finite group interval lattice. In particular we would like to show that the following question has a positive answer.

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Normal subsystems of fusion systems

2008-03-21

Michael Aschbacher

On intervals in subgroup lattices of finite groups

2008-03-17

Michael Aschbacher

We investigate the question of which finite lattices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic to the lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … We investigate the question of which finite lattices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic to the lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper H comma upper G right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[H,G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of all overgroups of a subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in 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>. We show that the structure 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> is highly restricted if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper H comma upper G right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[H,G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is disconnected. We define the notion of a “signalizer lattice" in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and show for suitable disconnected lattices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper H comma upper G right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[H,G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is minimal subject to being isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or its dual, then either <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 almost simple or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a signalizer lattice isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or its dual. We use this theory to answer a question in functional analysis raised by Watatani.

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Elementary abelian 2-subgroups of Sidki-type in finite groups

2007-12-31

Michael Aschbacher , Robert M. Guralnick , Yoav Segev

Let G be a finite group. We say that a nontrivial elementary abelian 2-subgroup V of G is of Sidki-type in G , if for each involution i in G … Let G be a finite group. We say that a nontrivial elementary abelian 2-subgroup V of G is of Sidki-type in G , if for each involution i in G , C_V(i) ≠ 1 . A conjecture due to S. Sidki (J. Algebra 39, 1976) asserts that if V is of Sidki-type in G , then V ∩ O_2(G) ≠ 1 . In this paper we prove a stronger version of Sidki's conjecture. As part of the proof, we also establish weak versions of the saturation results of G. Seitz (Invent. Math. 141, 2000) for involutions in finite groups of Lie type in characteristic 2 . Seitz's results apply to elements of order p in groups of Lie type in characteristic p , but only when p is a good prime, and 2 is usually not a good prime.

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Isotopy and geotopy for ternary rings of projective planes

2006-12-09

Michael Aschbacher

Standard components of alternating type centralized by a 4-group

2006-12-09

Michael Aschbacher

The limitations of nice mutually unbiased bases

2006-07-10

Michael Aschbacher , Andrew M. Childs , Paweł Wocjan

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Projective planes, loops, and groups

2006-01-19

Michael Aschbacher

Finite Bruck loops

2005-09-22

Michael Aschbacher , Michael Kinyon , J. D. Phillips

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Highly complex proofs and implications of such proofs

2005-09-09

Michael Aschbacher

Conventional wisdom says the ideal proof should be short, simple, and elegant. However there are now examples of very long, complicated proofs, and as mathematics continues to mature, more examples … Conventional wisdom says the ideal proof should be short, simple, and elegant. However there are now examples of very long, complicated proofs, and as mathematics continues to mature, more examples are likely to appear. Such proofs raise various issues. For example it is impossible to write out a very long and complicated argument without error, so is such a 'proof' really a proof? What conditions make complex proofs necessary, possible, and of interest? Is the mathematics involved in dealing with information rich problems qualitatively different from more traditional mathematics?

Coauthor	Papers Together
Stephen M. Smith	10
Stephen D. Smith	10
Yoav Segev	8
Ronald Solomon	5
Robert M. Guralnick	5
Richard Lyons	5
Bob Oliver	4
Michael Kinyon	3
Radha Kessar	3
Gary M. Seitz	3
Arjeh M. Cohen	2
William M. Kantor	2
Richard K. Lyons	2
Marshall Hall	2
Peter B. Kleidman	2
Daniel Gorenstein	2
J. D. Phillips	2
Andrew M. Childs	1
Dinakar Ramakrishnan	1
Olga Taussky	1
Finite Groups	1
Ulrich Meierfrankenfeld	1
Paweł Wocjan	1
J. D. Phillips	1
Richard M. Wilson	1
John Shareshian	1
Leonard L. Scott	1
Stephen Smith	1
Pierre Baldi	1
Andrew Chermak	1
Martin W. Liebeck	1
Bernd Stellmacher	1
Don Blasius	1
Eric B. Baum	1

Involutions in Chevalley groups over fields of even order

1976-10-01

Michael Aschbacher , Gary M. Seitz

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2-Fusion in Finite Groups

1974-01-01

David M. Goldschmidt

Suppose A c T c G are groups such that whenever a E A, g e G, and ag E T, then as e A. In this situation, we say … Suppose A c T c G are groups such that whenever a E A, g e G, and ag E T, then as e A. In this situation, we say that A is strongly closed in T with respect to G. We are concerned here with the case where G is finite, T is a Sylow 2-subgroup of G, and A is Abelian. In this case, we call G an S(A)-group. Our objective is to determine the composition factors of AG, the normal closure of A in G, when G is an S(A)-group. For this purpose, we make the following definitions: Suppose L is a perfect group such that L/Z(L) is simple. (Such a group is called quasi-simple.) We shall say that L is of type I provided L/Z(L) is isomorphic to one of the following: (a) L2(2), n > 3. (b) Sz(22n+l), n~ > 1. (c) U3 (2 ), n > 2. We shall say that L is of type II provided Z(L) has odd order and L = L/Z(L) satisfies one of the following conditions: (d) L L2(q), q 3, 5 (mod 8). (e) L is simple and contains an involution t such that (1) I L: CL(t) I is odd, (2) CL(t) = x K, (3) K contains a normal subgroup E of odd index such that CK(E) = 1 and E L2(q)g q _ 3, 5 (mod 8).

A Characterization of Chevalley Groups Over Fields of Odd Order

1977-11-01

Michael Aschbacher

Maximal subgroups of finite groups

1985-01-01

Michael Aschbacher , Leonard L. Scott

A Characterization of Chevalley Groups Over Fields of Odd Order

1977-09-01

Michael Aschbacher

The Classification of the Finite Simple Groups

1994-11-18

Daniel Gorenstein , Richard Lyons , Ronald Solomon

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 … 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.

Nonsolvable finite groups all of whose local subgroups are solvable

1968-01-01

John G. Thompson

Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups … Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups of symplectic type 397 5.4.^-groups, ^-solvability and F((&) 400 5.5.Groups of low order 404 5.6.2-groups, involutions and 2-length 409 5.7.Factorizations 423 5.8.Miscellaneous 426 6.A transitivity theorem 428 i 9 68]

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Finite groups generated by 3-transpositions. I

1971-09-01

Bernd Fischer

On finite groups of component type

1975-03-01

Michael Aschbacher

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Overgroups of Sylow subgroups in sporadic groups

1986-01-01

Michael Aschbacher

Central elements in core-free groups

1966-11-01

George Glauberman

On the maximal subgroups of the finite classical groups

1984-10-01

Michael Aschbacher

On Fusion Systems of Component Type

2019-01-01

Michael Aschbacher

The generalized Fitting subsystem of a fusion system

2010-12-20

Michael Aschbacher

Generation of fusion systems of characteristic 2-type

2009-12-10

Michael Aschbacher

Frobenius Categories versus Brauer Blocks

2009-01-01

Lluís Puig

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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Fusion systems

2016-06-29

Michael Aschbacher , Bob Oliver

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Finite Groups in Which the Centralizer of Any Element of Order 2 is 2-Closed

1965-07-01

Michio Suzuki

On finite groups generated by odd transpositions. I

1972-01-01

Michael Aschbacher

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.

Subgroup families controlling p-local finite groups

2005-08-23

Carles Broto , Natàlia Castellana , Jesper Grodal +2 more

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode 'conjugacy' relations among subgroups of S, and which are modelled on the … A p-local finite group consists of a finite p-group S, together with a pair of categories which encode 'conjugacy' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F-centric F-radical subgroups (at a minimum) to the set of F-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups. 2000 Mathematics Subject Classification 20J99 (primary), 55R35, 20D20 (secondary).

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The homotopy theory of fusion systems

2003-07-21

Carles Broto , Ran Levi , Bob Oliver

We define and characterize a class of $p$-complete spaces $X$ which have many of the same properties as the $p$-completions of classifying spaces of finite groups. For example, each such … We define and characterize a class of $p$-complete spaces $X$ which have many of the same properties as the $p$-completions of classifying spaces of finite groups. For example, each such $X$ has a Sylow subgroup $BS\longrightarrow X$, maps $BQ\longrightarrow X$ for a $p$-group $Q$ are described via homomorphisms $Q\longrightarrow S$, and $H^*(X;\mathbb {F}_p)$ is isomorphic to a certain ring of "stable elements" in $H^*(BS;\mathbb {F}_p)$. These spaces arise as the "classifying spaces" of certain algebraic objects which we call "$p$-local finite groups". Such an object consists of a system of fusion data in $S$, as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

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ON THE 1-COHOMOLOGY OF FINITE GROUPS OF LIE TYPE

1976-01-01

Wayne Jones , Brian Parshall

Extending Morphisms of Groups and Graphs

1992-03-01

Michael Aschbacher , Yoav Segev

Nonsolvable finite groups all of whose local subgroups are solvable. II

1970-05-01

John F. Thompson

In this second paper, the bulk of the work is devoted to characterizing E 2 (β) and S*(3).These two groups are "almost" ΛΓ-groups and it is relevant to treat them … In this second paper, the bulk of the work is devoted to characterizing E 2 (β) and S*(3).These two groups are "almost" ΛΓ-groups and it is relevant to treat them separately.The actual characterizations (Theorems 8.1 and 9.1) are very technical but the hypotheses deal with the structure and embedding in a simple group of certain {2, 3}-subgroups.This paper is a continuation of an earlier paper. 1 The bibliographical references are to I. 7* Groups in which 1 is the only p-signalizer* DEFINITION 7.1.%f*(p) = {S3 | (i) 35 is a subgroup of © of type (p, p).(ii) iV(S3) contains a S p -subgroup of ©.}.HYPOTHESIS 7.1.( i ) p is a prime and if S3e^*(p), then no i^-subgroup of C(S3) normalizes any nonidentity ^'-subgroup of ©. (ii) The centralizer of every nonidentity p-subgroup of © is psolvable.Lemmas 7.1, 7.2, 7.3 are proved under Hypothesis 7.1.LEMMA 7.1.(i) ^(ί))g^(p).(See Definitions 2.8 and 2.10 of I).(ii) If p^5, then <%f*(p) S gf (p).(iii) Ifp = S and if no element of ^(3) centralizes a quaternion subgroup of ©, then ^*(3)Sg'(3).Proof.If p is odd, choose S3e^*(p), while if p = 2, choose S3 e ^(2).We must show that either S3 centralizes every element of H(33; p') or p = 3, S3 e ^*(3) -^(3) and some element of ^(3) centralizes a quaternion subgroup of ©.Let ^ be a Sp-subgroup of JV(S3), so that $ is a S^-subgroup of {$.Proceeding by way of contradiction, let D be an element of M(S3; p') minimal subject to [O, S3] ^ 1.Then Q is a g-group for some prime q φ Pj Q = [£}, S3], and S3 0 = C^(Q) has order p.Let (£ -C(S3 0 ), e x = Cφ(S3 0 ), and let ^3* be a S^-subgroup of K containing (£ le Hypothesis 7.1 implies that O p/ (g) = 1.Let φ o = O p (<£).If [5β 0 , S3] S 33, then 1 Non-solvable finite groups all of whose local subgroups are solvable, I, Bull.Amer.Math.Soc.74 (1968), 383-437, which will be referred to as I. N-GROUPS II 457 33 centralizes O 2 (3i) and 31 is solvable.The proof is complete.We set ©! = {G I G e ©, C(G) is solvable.}& p = {G I G 6 ©, C(G) contains an elementary subgroup (£ of

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

1965-03-01

Daniel Gorenstein , John H. Walter

Signalizer lattices in finite groups

2009-05-01

Michael Aschbacher

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Finite permutation groups of rank 3

1964-01-01

D. G. Higman

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Transitive gruppen gerader ordnung, in denen jede involution genau einen punkt festläβt

1971-04-01

Helmut Bender

A classification of the maximal subgroups of the finite alternating and symmetric groups

1987-12-01

Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl

Centralizers of involutions in balanced groups

1972-02-01

Daniel Gorenstein , John H. Walter

On finite groups of Lie type and odd characteristic

1980-10-01

Michael Aschbacher

Normal subsystems of fusion systems

2008-03-21

Michael Aschbacher

Some subgroups of $SL_{n}(\mathbf{F}_{2})$

1969-03-01

J. E. McLaughlin

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On Representing Finite Lattices as Intervals in Subgroup Lattices of Finite Groups

1997-10-01

Robert W. Baddeley , Andrea Lucchini

On intervals in subgroup lattices of finite groups

2008-03-17

Michael Aschbacher

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Automorphisms of Trivalent Graphs

1980-03-01

David M. Goldschmidt

In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a is on adjacent vertices, then … In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a is on adjacent vertices, then its order divides 3 . 24. As observed by Sims [5], the hypothesis is equivalent to the following group-theoretic conditions: a) G is a finite group generated by a pair of subgroups {P1, Pd, b) i Pi: Pi n P2 1= 3 for i = 1, 2, c) no non-trivial normal subgroup of G is contained in P1 A, d) P1 and P2 are G-conjugate. What happens if we drop condition d) or, what is essentially the same thing, replace vertex transitive by edge transitive? This question is primarily motivated by the examples afforded by the rank 2 BN pairs over GF(2). In this case, the trivalent graph mentioned above is the so-called building associated to the BN pair [6]. In this paper, we classify all pairs of subgroups (P1, P2) for which hypotheses a), b) and c) are satisfied. There are precisely fifteen such pairs, and in particular, we find that P1 n P2 has order dividing 27. In order to describe the results more completely, let us define an amalgam to be a pair of group monomorphisms (5, 02) with the same domain: P, 1 B P2. We will say that (01, 02) is finite if both co-domains P1, P2 are finite. In this case, we define the index of the amalgam to be the pair of indices (I Pl: im s1 I, P2: im 02 1). By a completion of the amalgam we mean a pair of homomorphisms (*1, '2) to some group G making the obvious diagram commute, i.e., such that 01* = 102'2 (in right-hand notation). By abuse of notation, we may say that G is a completion of the amalgam. Of course, we always have the trivial completion g1 = g2 = 1We also always have the universal completion, usually known as the amalgamated product, from

On groups with a standard component of known type. II

1981-01-01

Michael Aschbacher , Gary M. Seitz

The Classification of the Finite Simple Groups

1980-06-01

Michael Aschbacher

�l�ments unipotents et sous-groupes paraboliques de groupes r�ductifs. I

1971-06-01

Armand Borel , Jacques Tits

Finite groups with a proper 2-generated core

1974-01-01

Michael Aschbacher

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A Local Approach to Buildings

1981-01-01

Jacques Tits

Component type theorems for finite groups in characteristic 2

1982-03-01

Richard Foote

< 2 in T centralizes /7(J), contrary to T/Oz(J) acting faithfully on/7(J < 2 in T centralizes /7(J), contrary to T/Oz(J) acting faithfully on/7(J

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Topology of order complexes of intervals in subgroup lattices

2003-07-16

John Shareshian

Flag structures on tits geometries

1983-03-01

Michael Aschbacher

Factorizations in Local Subgroups of Finite Groups

1977-12-31

George Glauberman

Reductions to local subgroups and sections Factorizations for $p=2$ The general situation Appendix A1. Proof of Theorem A Appendix A2. Corrections and additions to GL Bibliography. Reductions to local subgroups and sections Factorizations for $p=2$ The general situation Appendix A1. Proof of Theorem A Appendix A2. Corrections and additions to GL Bibliography.

A characterization of the unitary and symplectic groups over finite fields of characteristic at least 5

1973-07-01

Michael Aschbacher

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Simple connectivity ofp-group complexes

1993-06-01

Michael Aschbacher