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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 06 Dec 2023 14:11:23 +0000A characterization of the unitary and symplectic groups over finite fields of characteristic at least $5$
https://resolver.caltech.edu/CaltechAUTHORS:ASCpjm73
Authors: Aschbacher, Michael
Year: 1973
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 Zp x Zp, L2(pm) or SL2(pm), where m depends on A and B. Assume G has no nontrivial solvable normal subgroup. Then G is isomorphic to Spn(q) or Un(q) for some power q of p.https://authors.library.caltech.edu/records/679ps-2b104Classification of the Finite Simple Groups
https://resolver.caltech.edu/CaltechAUTHORS:20190814-100802612
Authors: Aschbacher, Michael
Year: 1980
DOI: 10.1007/bf03022850
The classification of the finite simple groups was completed sometime during the summer of 1980. To the extent that I can reconstruct things, the last piece in the puzzle was filled in by Ronald Solomon of Ohio State University. At the other chronological extreme, the theory of finite groups can be traced back to its beginnings in the early nineteenth century in the work of Abel, Cauchy, and Galois. Hence the problem of classifying the finite simple groups has a history of over a century and a half. The proof of the Classification Theorem is made up of thousands of pages in various mathematical journals with at least another thousand pages still left to appear in print. Many mathematicians have contributed to the proof; some have spent their entire mathematical lives working
on the problem.https://authors.library.caltech.edu/records/bh4sn-ayp61Finite simple groups and their subgroups
https://resolver.caltech.edu/CaltechAUTHORS:20201001-145810955
Authors: Aschbacher, Michael
Year: 1986
DOI: 10.1007/bfb0076170
The material in this article corresponds roughly to the contents of six lectures given at the International Symposium on Group Theory at Peking University in September 1984.
In essence the article describes the beginnings of a theory of permutation representations of finite groups based on the classifications of the finite simple groups. Chapter 3 is devoted to an outline of the Classification, with emphasis on recent efforts to improve the proof of the Classification Theorem. Chapters, 1, 2, and 6 discuss finite groups themselves, a notion of geometry due to J. Tits, and a class of group theoretical techniques introduced by B. Fischer. Each of these topics plays a role in the theory of permutation representations under discussion. The heart of the theory is the study of the subgroup structure of the finite simple groups. Certain results on this structure are described in Chapter 4 and 5.https://authors.library.caltech.edu/records/rddsk-b2514Geometries and Groups
https://resolver.caltech.edu/CaltechAUTHORS:20190814-100149802
Authors: Aschbacher, M.; Cohen, A. M.; Kantor, W. M.
Year: 1987
DOI: 10.1007/978-94-009-4017-8
The workshop was set up in order to stimulate the interaction between (finite and algebraic) geometries and groups. Five areas of concentrated research were chosen on which attention would be focused, namely: diagram geometries and chamber systems with transitive automorphism groups, geometries viewed as incidence systems, properties of finite groups of Lie type, geometries related to finite simple groups, and algebraic groups. The list of talks (cf. page iii) illustrates how these subjects were represented during the workshop. The contributions to these proceedings mainly belong to the first three areas; therefore, (i) diagram geometries and chamber systems with transitive automorphism groups, (ii) geometries viewed as incidence systems, and (iii) properties of finite groups of Lie type occur as section titles. The fourth and final section of these proceedings has been named graphs and groups; besides some graph theory, this encapsules most of the work related to finite simple groups that does not (explicitly) deal with diagram geometry. A few more words about the content: (i). Diagram geometries and chamber systems with transitive automorphism groups. As a consequence of Tits' seminal work on the subject, all finite buildings are known. But usually, in a situation where groups are to be characterized by certain data concerning subgroups, a lot less is known than the full parabolic picture corresponding to the building.https://authors.library.caltech.edu/records/my4ef-an845Some Multilinear Forms with Large Isometry Groups
https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055093
Authors: Aschbacher, Michael
Year: 1988
DOI: 10.1007/978-94-009-4017-8_15
Many groups are best described as the group of automorphisms of some natural object. I'm interested in obtaining such descriptions of the finite simple groups, and more generally descriptions of the groups of Lie type over arbitrary fields. The representation of the alternating group of degree n as the group of automorphisms of a set of order n is an excellent example of such a description. The representation of the classical groups as the isometry groups of bilinear or sequilinear forms is another.https://authors.library.caltech.edu/records/efe7x-yvd92Some multilinear forms with large isometry groups
https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055206
Authors: Aschbacher, Michael
Year: 1988
DOI: 10.1007/bf00191936
Many groups are best described as the group of automorphisms of some natural object. I'm interested in obtaining such descriptions of the finite simple groups, and more generally descriptions of the groups of Lie type over arbitrary fields. The representation of the alternating group of degree n as the group of automorphisms of a set of order n
is an excellent example of such a description. The representation of the classical groups as the isometry groups of bilinear or sequilinear forms is another.https://authors.library.caltech.edu/records/sfx6b-9r313The existence of J₃ and its embeddings in E₆
https://resolver.caltech.edu/CaltechAUTHORS:20200512-071755284
Authors: Aschbacher, Michael
Year: 1990
DOI: 10.1007/bf00147344
We determine the embeddings of the third sporadic group J₃ of Janko in simple Chevalley groups of type E₆ over finite and algebraically closed fields. As a corollary we obtain a short elegant existence proof of J₃. This is of interest as J₃ is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs were highly computational; cf. [4] and [6].https://authors.library.caltech.edu/records/pgbqn-knm13On a conjecture of Quillen and a lemma of Robinson
https://resolver.caltech.edu/CaltechAUTHORS:20200512-073808676
Authors: Aschbacher, Michael; Kleidman, Peter B.
Year: 1990
DOI: 10.1007/bf01191159
In this note we are concerned with finite groups G and primes p satisfying the Robinson Properties.https://authors.library.caltech.edu/records/dpjep-vcf30On conjectures of Guralnick and Thompson
https://resolver.caltech.edu/CaltechAUTHORS:20170810-072647371
Authors: Aschbacher, Michael
Year: 1990
DOI: 10.1016/0021-8693(90)90292-V
Given a permutation s on a finite set Ω of order n, define c(s) to be the number of cycles of sand Ind(s) = n - c(s).
Define a genus g system to be a triple ( G, Ω, S), where Ω is a finite set, G is a transitive subgroup of Sym(Ω), and S = (g_j: 1 ⩽j⩽r is a family of elements of G^# such that G = ⟨S⟩, g_1...g_r = 1, and 2(❘Ω❘ + g-1)= ∑_(j=1) Ind(g_j).https://authors.library.caltech.edu/records/tzbk6-99d36The uniqueness of groups of type J₄
https://resolver.caltech.edu/CaltechAUTHORS:20200512-075747975
Authors: Aschbacher, Michael; Segev, Yoav
Year: 1991
DOI: 10.1007/bf01232280
We give the first computer free proof of the uniqueness of groups of type J₄. In addition we supply simplified proofs of some properties of such groups, such as the structure of certain subgroups.
A group of type J₄ is a finite group G possessing an involution z such that H=C_G(z) satisfies F*(H)=Q is extraspecial of order 2¹³, H/Q is isomorphic to Z₃ extended by Aut (M₂₂), and z^G ⋂ Q ≠ {z}. We prove: Main Theorem. Up to isomorphism there exists at most one group of type J₄.https://authors.library.caltech.edu/records/pbths-spv27Exponents of almost simple groups and an application to the restricted Burnside problem
https://resolver.caltech.edu/CaltechAUTHORS:20200512-074240672
Authors: Aschbacher, Michael; Kleidman, Peter B.; Liebeck, Martin W.
Year: 1991
DOI: 10.1007/bf02571536
This paper is motivated by: The Restricted Burnside Problem R(n). For each r, are there only finitely many r-generator finite groups of exponent n?https://authors.library.caltech.edu/records/9835e-b1834Locally connected simplicial maps
https://resolver.caltech.edu/CaltechAUTHORS:20200512-125900851
Authors: Aschbacher, M.; Segev, Y.
Year: 1992
DOI: 10.1007/bf02773693
In Propositions 1.6 and 7.6 of his paper onp-group complexes of finite groups [5], Quillen establishes fundamental results comparing the homology and the fundamental group of the order complexes of posetsP, Q admitting a mapf :P →Q of posets with good local behavior. We prove the analogue of Quillen's results for mapsf :K→L of simplicial complexesK andL in a more general setup.https://authors.library.caltech.edu/records/0xgqd-p8x79Simple connectivity of p-group complexes
https://resolver.caltech.edu/CaltechAUTHORS:20200512-131404038
Authors: Aschbacher, Michael
Year: 1993
DOI: 10.1007/bf02808107
We investigate the simple connectivity ofp-subgroup complexes of finite groups.https://authors.library.caltech.edu/records/x9yyz-sa445Finite groups acting on homology manifolds
https://resolver.caltech.edu/CaltechAUTHORS:ASCpjm97
Authors: Aschbacher, Michael
Year: 1997
In this paper we study homology manifolds T admitting the action of a finite group preserving the structure of a regular CW-complex on T. The CW-complex is parameterized by a poset and the topological properties of the manifold are translated into a combinatorial setting via the poset. We concentrate on n-manifolds which admit a fairly rigid group of automorphisms transitive on the n-cells of the complex. This allows us to make yet another translation from a combinatorial into a group theoretic setting. We close by using our machinery to construct representations on manifolds of the Monster, the largest sporadic group. Some of these manifolds are of dimension 24, and hence candidates for examples to Hirzebruch's Prize Question in [HBJ], but unfortunately closer inspection shows the A^-genus of these manifolds is 0 rather than 1, so none is a Hirzebruch manifold.https://authors.library.caltech.edu/records/gp5x2-23y78Quasithin Groups
https://resolver.caltech.edu/CaltechAUTHORS:20200512-081608756
Authors: Aschbacher, Michael
Year: 1998
DOI: 10.1007/978-94-011-5308-9_18
The treatment of quasithin groups of characteristic 2 was one of the last steps in the Classification of the finite simple groups. Geoff Mason [12] announced a classification of these groups in about 1980, but never published his work. A few people have a copy of a large manuscript containing his efforts, but because it was distributed slowly, section by section, it was only during the last few years that it was realized that Mason's manuscript is incomplete in various ways. A few years ago I wrote up a treatment which begins where Mason's manuscript ends and finishes the problem assuming the results he says he proves. I have only read Mason's manuscript superficially, but it appears there are missing lemmas even for the part of the problem the theorems in his manuscript cover. I do believe however that he has seriously addressed the issues involved and that he could turn his manuscript into a proof with enough work. However Mason is now involved with Moonshine and has no interest in completing or publishing his manuscript.https://authors.library.caltech.edu/records/g57yg-2jx07Quasithin Groups
https://resolver.caltech.edu/CaltechAUTHORS:20200512-100004286
Authors: Aschbacher, Michael G.; Smith, Stephen D.
Year: 1998
DOI: 10.1007/978-3-0348-8819-6_1
Geoff Mason announced in about 1980 the classifcication of quasithin groups of characteristic 2; but never published this step in the classification of the finite simple groups. In January 1996, the authors began work toward a new and more general classification of quasithin groups; the paper gives an exposition of the approach and considerable progress to date.https://authors.library.caltech.edu/records/ytern-gd019On Primitive Linear Representations of Finite Groups
https://resolver.caltech.edu/CaltechAUTHORS:20170710-101117286
Authors: Aschbacher, Michael
Year: 2000
DOI: 10.1006/jabr.2000.8532
Let F be a field, let G be a finite group, and let π be a linear representation of G over F; that is, π is a group homomorphism π: G → GL(V) of G into the general linear group on a finite-dimensional vector space V over F.
We say π is AI if π is completely reducible and for each normal subgroup H of G, each irreducible FH-submodule of V is absolutely irreducible. For example, if F is algebraically closed then all completely reducible representations over F are AI. In particular, all of our theorems hold over the complex numbers without the hypothesis that the representation is AI.https://authors.library.caltech.edu/records/n5tc0-76369A 2-local characterization of M(12)
https://resolver.caltech.edu/CaltechAUTHORS:ASCijm03
Authors: Aschbacher, Michael
Year: 2003
A characterization of the Mathieu group M(12) is established; the characterization is used by Aschbacher and Smith in their classification of the quasithin finite simple groups.https://authors.library.caltech.edu/records/m3wrn-5w840The Status of the Classification of the Finite Simple Groups
https://resolver.caltech.edu/CaltechAUTHORS:20110817-103540228
Authors: Aschbacher, Michael
Year: 2004
The classification of the finite simple groups is one of the great theorems of recent mathematics. One of its principal participants reviews the result and current progress on understanding it.https://authors.library.caltech.edu/records/5b4mm-frv50Highly complex proofs and implications of such proofs
https://resolver.caltech.edu/CaltechAUTHORS:20200916-090615838
Authors: Aschbacher, Michael
Year: 2005
DOI: 10.1098/rsta.2005.1655
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?https://authors.library.caltech.edu/records/38qww-ejv63Finite Bruck loops
https://resolver.caltech.edu/CaltechAUTHORS:20110209-094820472
Authors: Aschbacher, Michael; Kinyon, Michael K.; Phillips, J. D.
Year: 2006
DOI: 10.1090/S0002-9947-05-03778-5
Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops X, showing that X is essentially the direct product of a Bruck loop of odd order with a 2-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite 2-element Bruck loops are 2-loops, leaving open the question of whether such obstructions actually exist.https://authors.library.caltech.edu/records/c4adw-9py07Elementary abelian 2-subgroups of Sidki-type in finite groups
https://resolver.caltech.edu/CaltechAUTHORS:20100503-094555816
Authors: Aschbacher, Michael; Guralnick, Robert; Segev, Yoav
Year: 2007
DOI: 10.4171/GGD/18
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 ∩ 0_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.https://authors.library.caltech.edu/records/47bvp-rpt47The limitations of nice mutually unbiased bases
https://resolver.caltech.edu/CaltechAUTHORS:20190826-124740760
Authors: Aschbacher, Michael; Childs, Andrew M.; Wocjan, Paweł
Year: 2007
DOI: 10.1007/s10801-006-0002-y
Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets.https://authors.library.caltech.edu/records/h0k3e-2fp33On intervals in subgroup lattices of finite groups
https://resolver.caltech.edu/CaltechAUTHORS:ASCjams08
Authors: Aschbacher, Michael
Year: 2008
DOI: 10.1090/S0894-0347-08-00602-4
We investigate the question of which finite lattices L are isomorphic to the lattice [H,G] of all overgroups of a subgroup H in a finite group G. We show that the structure of G is highly restricted if [H,G] is disconnected. We define the notion of a "signalizer lattice" in H and show for suitable disconnected lattices L, if [H,G] is minimal subject to being isomorphic to L or its dual, then either G is almost simple or H admits a signalizer lattice isomorphic to L or its dual. We use this theory to answer a question in functional analysis raised by Watatani.https://authors.library.caltech.edu/records/cm5e5-ptt87Normal subsystems of fusion systems
https://resolver.caltech.edu/CaltechAUTHORS:20180810-075919663
Authors: Aschbacher, Michael
Year: 2008
DOI: 10.1112/plms/pdm057
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.https://authors.library.caltech.edu/records/s19mn-5r963On a question of Farjoun
https://resolver.caltech.edu/CaltechAUTHORS:20180802-143851994
Authors: Aschbacher, Michael
Year: 2008
DOI: 10.1515/9783110198126.1
[no abstract]https://authors.library.caltech.edu/records/gst1n-3qv43Signalizer lattices in finite groups
https://resolver.caltech.edu/CaltechAUTHORS:20090811-091248605
Authors: Aschbacher, Michael
Year: 2009
DOI: 10.1307/mmj/1242071684
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.https://authors.library.caltech.edu/records/p2hzh-8r107Overgroups of Primitive Groups
https://resolver.caltech.edu/CaltechAUTHORS:20091020-133518121
Authors: Aschbacher, Michael
Year: 2009
DOI: 10.1017/S1446788708000785
We give a qualitative description of the set O_G(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 O_G(H) impose strong restrictions on H and its overgroup lattice.https://authors.library.caltech.edu/records/3m2gh-ec774Overgroups of primitive groups, II
https://resolver.caltech.edu/CaltechAUTHORS:20090817-144817879
Authors: Aschbacher, Michael
Year: 2009
DOI: 10.1016/j.jalgebra.2009.04.044
We continue our study of the overgroup lattices of subgroups of finite alternating and symmetric groups, with applications to the question of Palfy and Pudlak as to whether each finite lattice is an interval in the lattice of subgroups of some finite group.https://authors.library.caltech.edu/records/7kxjk-xhy82Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups
https://resolver.caltech.edu/CaltechAUTHORS:20091210-094932243
Authors: Aschbacher, Michael; Shareshian, John
Year: 2009
DOI: 10.1016/j.jalgebra.2009.05.042
Let G be a finite alternating or symmetric group. We describe an infinite class of finite lattices, none of which is isomorphic to any interval [H,G] in the subgroup lattice of G.https://authors.library.caltech.edu/records/be14c-2nz37A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver
https://resolver.caltech.edu/CaltechAUTHORS:20100806-092914498
Authors: Aschbacher, Michael; Chermak, Andrew
Year: 2010
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 = (G_0 → G_1 → ...) of 2-local finite groups, via a representation of a chain G^* = (G_0 → G_1 → ...) of groups, such that G_0 is the 2-local finite group of the third Conway sporadic group Co_3, and for n > 0, G_n is one of the 2-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit G 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 G is shown to be the classifying space B DI(4) of the exotic 2-compact group of Dwyer and Wilkerson [DW93].https://authors.library.caltech.edu/records/qta7e-y5811Generation of fusion systems of characteristic 2-type
https://resolver.caltech.edu/CaltechAUTHORS:20100513-152930353
Authors: Aschbacher, Michael
Year: 2010
DOI: 10.1007/s00222-009-0229-z
We prove that if F is a saturated fusion system on a finite 2-
group S, then either F is known, or F is generated by the normalizers of
two canonically defined F-characteristic subgroups of S. There are various
corollaries for finite groups of characteristic 2-type.https://authors.library.caltech.edu/records/bgvds-kvv30Fusion Systems in Algebra and Topology
https://resolver.caltech.edu/CaltechAUTHORS:20180807-124803752
Authors: Aschbacher, Michael; Kessar, Radha; Oliver, Bob
Year: 2011
DOI: 10.1017/CBO9781139003841
A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.https://authors.library.caltech.edu/records/r5359-bqa87Lower signalizer lattices in alternating and symmetric groups
https://resolver.caltech.edu/CaltechAUTHORS:20120525-104153526
Authors: Aschbacher, Michael
Year: 2012
DOI: 10.1515/jgt-2011-0112
We prove that the subgroup lattices of finite alternating and symmetric groups
do not contain so-called lower signalizer lattices in the class D�Δ. This result is one step
in a program to show that the lattices in the class D�Δ are not isomorphic to an interval in
the subgroup lattice of any finite group.https://authors.library.caltech.edu/records/9saf3-dsg112012 Steele Prizes
https://resolver.caltech.edu/CaltechAUTHORS:20180807-133142929
Authors: Aschbacher, Michael; Lyons, Richard; Smith, Steve; Solomon, Ronald
Year: 2012
DOI: 10.1090/noti826
The 2012 Leroy P. Steele Prize for Mathematical
Exposition is awarded to Michael Aschbacher,
Richard Lyons, Steve Smith, and Ronald Solomon
for their work, The Classification of Finite Simple
Groups: Groups of Characteristic 2 Type, Mathematical
Surveys and Monographs, 172, American
Mathematical Society, Providence, RI, 2011. In this
paper, the authors, who have done foundational
work in the classification of finite simple groups,
offer to the general mathematical public an articulate
and readable exposition of the classification
of characteristic 2 type groups.https://authors.library.caltech.edu/records/td2h0-tae78S_3-free 2-fusion systems
https://resolver.caltech.edu/CaltechAUTHORS:20130607-083326320
Authors: Aschbacher, Michael
Year: 2013
DOI: 10.1017/S0013091512000235
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.https://authors.library.caltech.edu/records/tecsz-6kh56Fusion systems of F_2-type
https://resolver.caltech.edu/CaltechAUTHORS:20130327-114824399
Authors: Aschbacher, Michael
Year: 2013
DOI: 10.1016/j.jalgebra.2012.12.018
We prove results on 2-fusion systems related to the 2-fusion systems of groups of Lie type over the field of order 2 and certain sporadic groups. The results are used in a later paper to determine the N-systems: the 2-fusion systems of N-groups.https://authors.library.caltech.edu/records/npe69-kz005Overgroup lattices in finite groups of Lie type containing a parabolic
https://resolver.caltech.edu/CaltechAUTHORS:20130509-100832818
Authors: Aschbacher, Michael
Year: 2013
DOI: 10.1016/j.jalgebra.2013.01.034
The main theorem is a step in a program to show there exist finite lattices that are not an interval in the lattice of subgroups of any finite group. As part of the proof of the main theorem, we prove a theorem on the structure of maximal parabolics in finite groups of Lie type, which is of independent interest.https://authors.library.caltech.edu/records/1f057-24g21Finite Groups of Seitz Type
https://resolver.caltech.edu/CaltechAUTHORS:20131209-104216789
Authors: Aschbacher, Michael
Year: 2014
DOI: 10.1090/S0002-9939-2013-11752-1
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.https://authors.library.caltech.edu/records/5af6y-91a67Overgroups of Root Groups in Classical Groups
https://resolver.caltech.edu/CaltechAUTHORS:20160425-141046633
Authors: Aschbacher, Michael
Year: 2016
DOI: 10.1090/memo/1140
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.https://authors.library.caltech.edu/records/yzaqr-6fd02Daniel Gorenstein, 1923-1992 - A Biographical Memoir by
Michael Aschbacher
https://resolver.caltech.edu/CaltechAUTHORS:20160708-073018195
Authors: Aschbacher, Michael
Year: 2016
Daniel Gorenstein was one of the most influential figures
in mathematics during the last few decades of the 20th
century. In particular, he was a primary architect of the
classification of the finite simple groups.
During his career Gorenstein received many of the honors
that the mathematical community reserves for its highest
achievers. He was awarded the Steele Prize for mathematical
exposition by the American Mathematical Society in
1989; he delivered the plenary address at the International
Congress of Mathematicians in Helsinki, Finland, in 1978;
and he was the Colloquium Lecturer for the American
Mathematical Society in 1984. He was also a member of
the National Academy of Sciences and of the American
Academy of Arts and Sciences.
Gorenstein was the Jacqueline B. Lewis Professor of
Mathematics at Rutgers University and the founding director of its Center for Discrete
Mathematics and Theoretical Computer Science. He served as chairman of the university's
mathematics department from 1975 to 1982, and together with his predecessor, Ken
Wolfson, he oversaw a dramatic improvement in the quality of mathematics at Rutgers.https://authors.library.caltech.edu/records/1dh85-bt711N-groups and fusion systems
https://resolver.caltech.edu/CaltechAUTHORS:20160527-090500481
Authors: Aschbacher, Michael
Year: 2016
DOI: 10.1016/j.jalgebra.2015.10.011
We classify all saturated 2-fusion systems that are N-systems: that is those systems all of whose local subsystems are solvable, subject to one of the two possible notions of solvability. We also use the result on fusion systems to give a new proof of Thompson's theorem on N-groups; indeed we give a new proof of the theorem determining all finite groups in which all 2-locals are solvable.https://authors.library.caltech.edu/records/twt7w-tq484Fusion systems
https://resolver.caltech.edu/CaltechAUTHORS:20161116-124215563
Authors: Aschbacher, Michael; Oliver, Bob
Year: 2016
DOI: 10.1090/bull/1538
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.https://authors.library.caltech.edu/records/dfevp-kh744The subgroup structure of finite groups
https://resolver.caltech.edu/CaltechAUTHORS:20170928-081918166
Authors: Aschbacher, Michael
Year: 2017
DOI: 10.1090/conm/694/13959
[No abstract]https://authors.library.caltech.edu/records/hv2fa-w5m79The 2-fusion system of an almost simple group
https://resolver.caltech.edu/CaltechAUTHORS:20190822-100100397
Authors: Aschbacher, Michael
Year: 2020
DOI: 10.1016/j.jalgebra.2019.08.017
We show that, under suitable local constraints, the 2-fusion system of an almost simple finite group is almost simple.https://authors.library.caltech.edu/records/c2t20-f0d81Fusion systems with alternating J‐components
https://resolver.caltech.edu/CaltechAUTHORS:20200515-104035073
Authors: Aschbacher, Michael
Year: 2020
DOI: 10.1112/jlms.12335
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.https://authors.library.caltech.edu/records/cny0p-qk635Walter's basic theorem for fusion systems
https://resolver.caltech.edu/CaltechAUTHORS:20201217-124921798
Authors: Aschbacher, Michael
Year: 2021
DOI: 10.1016/j.jalgebra.2020.11.018
This is the first of two papers determining the saturated 2-fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2-fusion system of a large group of Lie type over a field of odd order.https://authors.library.caltech.edu/records/2ayzz-p1011Walter's theorem for fusion systems
https://resolver.caltech.edu/CaltechAUTHORS:20201023-130934536
Authors: Aschbacher, Michael
Year: 2021
DOI: 10.1112/plms.12386
We determine the saturated 2‐fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2‐fusion system of a large group of Lie type over a field of odd order.https://authors.library.caltech.edu/records/fbad7-znn76Fusion systems with J-components over F-_(2^e) with e > 1
https://resolver.caltech.edu/CaltechAUTHORS:20220901-221643366
Authors: Aschbacher, Michael
Year: 2022
DOI: 10.1515/jgth-2020-0156
Let κ be a finite simple group of Lie type over a field of even order q > 2. If κ is not ²F₄(q), then we determine the fusion systems ℱ of J-component type with a fully centralized involution j such that C_(ℱ)(j) has a component realized by κ. The exceptional case is treated in a later paper.https://authors.library.caltech.edu/records/k0xg6-g5870Fusion systems with U₃(3) J-components
https://resolver.caltech.edu/CaltechAUTHORS:20210610-093254072
Authors: Aschbacher, Michael
Year: 2022
DOI: 10.1016/j.jalgebra.2021.06.006
We determine the 2-fusion systems of J-component type in which the centralizer of some fully centralized involution has a maximal J-component that is the 2-fusion system of U₃(3).https://authors.library.caltech.edu/records/vtna3-xc530Fusion systems with 2-small components
https://resolver.caltech.edu/CaltechAUTHORS:20230725-746861000.32
Authors: Aschbacher, Michael
Year: 2023
DOI: 10.1090/tran/8797
We show that there are no odd simple 2-fusion systems F in which the centralizer of some fully centralized involution contains a component C that is the 2-fusion system of a simple group K such that C is J-maximal or maximal and subintrinsic in C(F), as appropriate, and such that K is of Lie type over the field of order 2, but not Spₙ(2) or F₄(2); or K is one of many sporadic groups; or K is PΩ⁺₈(3).https://authors.library.caltech.edu/records/ne4cd-2a765