Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type

176 Pages·2019·2.439 MB·English

by  Jesper M. Moller, Bob Oliver, Carles Broto

Number 1267
Automorphisms of Fusion Systems
of Finite Simple Groups of Lie Type
Carles Broto
Jesper M. Møller
Bob Oliver
Automorphisms of Fusion Systems
of Sporadic Simple Groups
Bob Oliver
November 2019 • Volume 262 • Number 1267 (fourth of 7 numbers)
Number 1267
Automorphisms of Fusion Systems
of Finite Simple Groups of Lie Type
Carles Broto
Jesper M. Møller
Bob Oliver
Automorphisms of Fusion Systems
of Sporadic Simple Groups
Bob Oliver
November 2019 • Volume 262 • Number 1267 (fourth of 7 numbers)
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DOI:https://doi.org/10.1090/memo/1267
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Contents
Automorphisms of Fusion Systems of Finite Simple Groups
of Lie Type
by Carles Broto, Jesper M. Møller, and Bob Oliver 1
Introduction 3
Tables of substitutions for Theorem B 8
Chapter 1. Tame and reduced fusion systems 13
Chapter 2. Background on finite groups of Lie type 23
Chapter 3. Automorphisms of groups of Lie type 31
Chapter 4. The equicharacteristic case 37
Chapter 5. The cross characteristic case: I 51
Chapter 6. The cross characteristic case: II 79
Appendix A. Injectivity of μ by Bob Oliver 93
G
A.1. Classical groups of Lie type in odd characteristic 96
A.2. Exceptional groups of Lie type in odd characteristic 98
Bibliography for Automorphisms of Fusion Systems of Finite Simple Groups of
Lie Type 115
Automorphisms of Fusion Systems of Sporadic Simple Groups
by Bob Oliver 119
Introduction 121
Chapter 1. Automorphism groups of fusion systems: Generalities 127
Chapter 2. Automorphisms of 2-fusion systems of sporadic groups 131
Chapter 3. Tameness at odd primes 137
Chapter 4. Tools for comparing automorphisms of fusion and linking systems145
Chapter 5. Injectivity of μ 151
G
Bibliography for Automorphisms of Fusion Systems of Sporadic Simple Groups161
iii

Abstract
Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type
by Carles Broto, Jesper M. Møller, and Bob Oliver
For a finite group G of Lie type and a prime p, we compare the automorphism
groupsofthefusionandlinkingsystemsofGatpwiththeautomorphism groupof
G itself. When p is the defining characteristic of G, they are all isomorphic, with a
very short list of exceptions. When p is different from the defining characteristic,
thesituationismuchmorecomplex, butcanalwaysbereducedtoacasewherethe
natural map from Out(G) to outer automorphisms of the fusion or linking system
is split surjective. This work is motivated in part by questions involving extending
thelocalstructureofagroupbyagroupofautomorphisms, andinpartbywanting
to describe self homotopy equivalences of BG∧ in terms of Out(G).
p
ReceivedbytheeditorJanuary18,2016and,inrevisedform,January27,2017andFebruary
1,2017.
ArticleelectronicallypublishedonDecember27,2019.
DOI:https://doi.org/10.1090/memo/1267
2010MathematicsSubjectClassification. [AutomorphismsofFusionSystemsofFiniteSimple
GroupsofLieType]Primary20D06. Secondary20D20,20D45,20E42,55R35;[Automorphismsof
FusionSystemsofSporadicSimple Groups]Primary: 20E25,20D08. Secondary: 20D20,20D05,
20D45.
Key words and phrases. [AutomorphismsofFusionSystemsofFinite Simple Groups ofLie
Type]GroupsofLietype,fusionsystems,automorphisms,classifyingspaces;[Automorphismsof
Fusion Systems of Sporadic Simple Groups] Fusion systems, sporadic groups, Sylow subgroups,
finitesimplegroups.
C. Broto is partially supported by MICINN grant MTM2010-20692 and MINECO grant
MTM2013-42293-P. He is affiliated with the Departament de Matem`atiques, Universitat
Aut`onomadeBarcelona,E–08193Bellaterra,Spain. Email: [e-mail].
J. Møller is partially supported by the Danish National Research Foundation through the
Centre for Symmetry and Deformation (DNRF92) and by Villum Fonden through the project
ExperimentalMathematicsinNumberTheory,OperatorAlgebras,andTopology. Heisaffiliated
with the Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark. Email:
[e-mail].
B.OliverispartiallysupportedbyUMR7539oftheCNRS,andbyprojectANRBLAN08-
2 338236,HGRT.HeisaffiliatedwiththeUniversit´eParis13,SorbonneParisCit´e,LAGA,UMR
7539 du CNRS, 99, Av. J.-B. Cl´ement, 93430 Villetaneuse, France. Email: [e-mail]-
paris13.fr.
All three authors wish to thank Københavns Universitet, the Universitat Auto`noma de
Barcelona, and especially the Centre for Symmetry and Deformation in Copenhagen, for their
hospitalitywhilemuchofthisworkwascarriedout.
(cid:2)c2019 American Mathematical Society
v
vi ABSTRACT
Automorphisms of Fusion Systems of Sporadic Simple Groups
by Bob Oliver
We prove here that with a very small number of exceptions, when G is a
sporadic simple group and p is a prime such that the Sylow p-subgroups of G are
nonabelian, then Out(G) is isomorphic to the outer automorphism groups of the
fusion and linking systems of G. In particular, the p-fusion system of G is tame in
∼
the sense of [AOV1], and is tamely realized by G itself except when G=M and
11
p = 2. From the point of view of homotopy theory, these results also imply that
Out(G)∼=Out(BG∧) in many (but not all) cases.
p
Automorphisms of Fusion Systems
of Finite Simple Groups
of Lie Type
by Carles Broto, Jesper M. Møller, and Bob Oliver

Introduction
When p is a prime, G is a finite group, and S ∈ Syl (G), the fusion system
p
of G at S is the category F (G) whose objects are the subgroups of S, and whose
S
morphisms are those homomorphisms between subgroups induced by conjugation
in G. In this paper, we are interested in comparing automorphisms of G, when
G is a simple group of Lie type, with those of the fusion system of G at a Sylow
p-subgroup of G (for different primes p).
Rather than work withautomorphisms ofF (G)itself, it turns out tobe more
S
naturalinmanysituationstostudythegroupOut(Lc(G))ofouterautomorphisms
S
ofthecentriclinkingsystem ofG. WerefertoChapter1forthedefinitionofLc(G),
S
andtoDefinition1.2forprecisedefinitionsofOut(F (G))andOut(Lc(G)). These
S S
are defined in such a way that there are natural homomorphisms
Out(G)−−−κ−G−→Out(LcS(G))−−−μ−G−→Out(FS(G)) and κG =μG◦κG.
For example, if S controls fusion in G (i.e., if S has a normal complement), then
Out(F (G)) = Out(S), and κ is induced by projection to S. The fusion system
S G
F (G) is tamely realized by G if κ is split surjective, and is tame if it is tamely
S G
realizedbysomefinitegroupG∗ whereS ∈Syl (G∗)andF (G)=F (G∗). Tame-
p S S
ness plays an important role in Aschbacher’s program for shortening parts of the
proofoftheclassificationoffinitesimplegroupsbyclassifyingsimplefusionsystems
over finite 2-groups. We say more about this later in the introduction, just before
the statement of Theorem C.
By [BLO1, Theorem B], Out(Lc(G)) ∼= Out(BG∧): the group of homotopy
S p
classes of self homotopy equivalences of the p-completed classifying space of G.
Thus one of the motivations for this paper is to compute Out(BG∧) when G is a
p
finitesimplegroupofLietype(incharacteristicporincharacteristicdifferentfrom
p), and compare it with Out(G).
Following the notation used in [GLS3], for each prime p, we let Lie(p) denote
theclassoffinitegroupsofLietypeincharacteristicp,andletLiedenotetheunion
of the classes Lie(p) for all primes p. (See Definition 2.1 for the precise definition.)
We say that G ∈ Lie(p) is of adjoint type if Z(G) = 1, and is of universal type if
it has no nontrivial central extensions which are in Lie(p). For example, for n≥2
and q a power of p, PSL (q) is of adjoint type and SL (q) of universal type.
n n
Our results can be most simply stated in the “equi-characteristic case”: when
working with p-fusion of G∈Lie(p).
Theorem A. Let p be a prime. Assume that G ∈ Lie(p) and is of universal
or adjoint type, and also that (G,p) ∼=(cid:7) (Sz(2),2). Fix S ∈ Syl (G). Then the
p
composite homomorphism
κ : Out(G)−−−κ−G−−→Out(Lc(G))−−−μ−−G−→Out(F (G))
G S S
∼
is an isomorphism, and κ and μ are isomorphisms except when G=PSL (2).
G G 3
3