Table Of ContentLecture Notes in Mathematics 2302
Nick Gill
Martin W. Liebeck
Pablo Spiga
Cherlin’s Conjecture
for Finite Primitive
Binary Permutation
Groups
Lecture Notes in Mathematics
Volume 2302
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
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Nick Gill (cid:129) Martin W. Liebeck (cid:129) Pablo Spiga
Cherlin’s Conjecture for
Finite Primitive Binary
Permutation Groups
NickGill MartinW.Liebeck
SchoolofMathematicsandStatistics DepartmentofMathematics
TheOpenUniversity ImperialCollegeLondon
MiltonKeynes,UK London,UK
PabloSpiga
DipartimentodiMatematicapurae
Applicata
UniversityofMilano-Bicocca
Milano,Italy
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-95955-5 ISBN978-3-030-95956-2 (eBook)
https://doi.org/10.1007/978-3-030-95956-2
Funding Information: Engineering and Physical Sciences Research Council EP/R028702/1
(http://dx.doi.org/10.13039/501100000266)
MathematicsSubjectClassification:20B05,20D06,03C13
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland
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Contents
1 Introduction .................................................................. 1
1.1 Basics:TheDefinitionofRelationalComplexity.................... 4
1.1.1 RelationalStructures.......................................... 4
1.1.2 Tuples.......................................................... 7
1.2 Basics:SomeKeyExamples.......................................... 9
1.2.1 ExistingResultsonRelationalComplexity.................. 13
1.3 Motivation:OnHomogeneity......................................... 15
1.3.1 ExistingResultsonHomogeneity............................ 16
1.4 Motivation:OnModelTheory........................................ 19
1.5 Motivation:OtherImportantStatistics ............................... 20
1.6 BeyondtheBinaryConjecture........................................ 23
1.7 Methods:BasicLemmas.............................................. 24
1.7.1 RelationalComplexityandSubgroups....................... 25
1.7.2 RelationalComplexityandSubsets .......................... 26
1.7.3 StronglyNon-binarySubsets ................................. 27
1.8 Methods:FrobeniusGroups........................................... 30
1.9 Methods:OnComputation............................................ 33
2 PreliminaryResultsforGroupsofLieType.............................. 37
2.1 ResultsonAlternatingSections....................................... 38
2.2 StabilizersContainingCertainElements............................. 40
2.2.1 SomeGroupsthatAreNotAlmostSimple .................. 41
2.2.2 ClassicalGroups............................................... 42
2.2.3 ExceptionalGroups ........................................... 59
2.3 ResultsonOdd-DegreeActions ...................................... 62
2.4 ResultsonCentralizers................................................ 67
2.5 OuterAutomorphismsofGroupsofLieType ....................... 67
2.6 OnFusionandFactorization.......................................... 68
3 ExceptionalGroups.......................................................... 71
3.1 MaximalSubgroupsofExceptionalGroupsofLieType............ 71
3.2 SmallExceptionalGroupsofLieType............................... 73
vii
viii Contents
3.3 MaximalSubgroupsin(I)ofTheorem3.1.1:Parabolic
Subgroups.............................................................. 76
3.4 Type(II),Part1:ReductiveSubgroupsofMaximalRank........... 79
3.5 Type(II),Part2:MaximalTorusNormalizers ....................... 86
3.6 MaximalSubgroupsin(V)ofTheorem3.1.1........................ 91
3.7 MaximalSubgroupsin(VI)ofTheorem3.1.1....................... 96
3.8 TheRemainingFamiliesinTheorem3.1.1........................... 98
3.8.1 Type(III)....................................................... 98
3.8.2 Type(IV) ...................................................... 99
3.8.3 Type(VII)...................................................... 99
3.8.4 Type(VIII)..................................................... 99
3.8.5 Type(IX) ...................................................... 100
3.8.6 Type(X) ....................................................... 100
4 ClassicalGroups ............................................................. 103
4.1 BackgroundonClassicalGroups..................................... 103
4.1.1 BasicAssumptions............................................ 104
4.2 FamilyC ............................................................... 105
2
4.2.1 CaseS =SL (q).............................................. 106
n
4.2.2 TheTotallySingularCase .................................... 109
4.2.3 AGeneralReduction.......................................... 112
4.2.4 CaseWhereS =SU (q)andtheW
n i
AreNon-degenerate........................................... 118
4.2.5 CaseWhereS =Sp (q)andtheW
n i
AreNon-degenerate........................................... 120
4.2.6 CaseWhereS = (cid:2) (q)fornq Odd,andtheW
n i
AreNon-degenerate........................................... 122
4.2.7 CaseWhereS =(cid:2)±(q)andtheW
n i
AreNon-degenerate........................................... 123
4.3 FamilyC ............................................................... 126
3
4.3.1 CaseS =SL (q).............................................. 127
n
4.3.2 CaseS =SU (q).............................................. 129
n
4.3.3 CaseS =Sp (q) .............................................. 133
n
4.3.4 CaseS =(cid:2)(cid:3)(q)............................................... 135
n
4.4 FamilyC ............................................................... 138
4
4.4.1 CaseS =SL (q).............................................. 139
n
4.4.2 CaseS =SU (q).............................................. 141
n
4.4.3 CaseWhereS IsSymplecticorOrthogonal ................. 143
4.4.4 TheRemainingCases......................................... 148
4.5 FamilyC ............................................................... 149
5
4.5.1 CaseS =SL (q).............................................. 149
n
4.5.2 CaseS =SU (q).............................................. 151
n
4.5.3 CaseS =Sp (q) .............................................. 156
n
4.5.4 CaseS IsOrthogonal.......................................... 157
4.6 FamilyC ............................................................... 158
6
Contents ix
4.7 FamilyC ............................................................... 164
7
4.7.1 CaseS =SL (q).............................................. 168
n
4.7.2 CaseS =SU (q).............................................. 170
n
4.7.3 CaseS =Sp (q) .............................................. 172
n
4.7.4 CaseS =(cid:2) (q),nOdd....................................... 175
n
4.7.5 CaseS =(cid:2)+(q) .............................................. 176
n
4.8 FamilyC ............................................................... 180
8
4.8.1 CaseS =SL (q).............................................. 180
n
4.8.2 CaseS =Sp (q) .............................................. 182
n
4.9 FamilyS ............................................................... 182
4.9.1 Strategies....................................................... 183
4.9.2 TheCaseWhereM IsAlternating........................... 187
0
4.9.3 TheCaseWhereM IsSporadic ............................. 194
0
4.9.4 TheCaseWhereM IsofLieType .......................... 194
0
4.10 ExceptionalAutomorphisms.......................................... 205
Bibliography...................................................................... 209
Chapter 1
Introduction
In this monograph, we are concerned with the problem of classifying the finite
primitive binary permutation groups. Let G be a permutationgroup on the set (cid:2).
Givenapositiveintegern,givenI :=(ω ,ω ,...,ω )intheCartesianproduct(cid:2)n
1 2 n
andgiveng ∈G,wewrite
Ig :=(ωg,ωg,...,ωg).
1 2 n
Moreover,for every 1 ≤ i < j ≤ n, we let I := (ω ,ω ) be the 2-subtupleof
ij i j
I correspondingtotheith andtothejth coordinate.Now,thepermutationgroupG
on(cid:2)iscalledbinaryif,forallpositiveintegersn,andforallI andJ in(cid:2)n,there
existsg ∈GsuchthatIg =J ifandonlyifforall2-subtuples,I ,ofI,thereexists
ij
anelementg suchthatIgij =J .
ij ij ij
Cherlinhasproposedaconjecturelistingthefiniteprimitivebinarypermutation
groups[21]. Theconjectureisasfollows,andourtask istocompletethe proofof
thisconjecture.
Conjecture1.1 A finite primitive binary permutation group must be one of the
following:
1. asymmetricgroupSym(n)actingnaturallyonnelements;
2. acyclicgroupofprimeorderactingregularlyonitself;
3. an affineorthogonalgroupV (cid:2)O(V) with V a vectorspaceovera finite field
equippedwith a non-degenerateanisotropicquadraticform, actingonitself by
translation,withcomplementthefullorthogonalgroupO(V).
TheterminologyofConjecture1.1isfullyexplainedinsubsequentsections.In
particular,wegivetwoequivalentdefinitionsoftheadjective“binary”inSect.1.1,
andallthreefamilieslistedinConjecture1.1arefullydiscussedinSect.1.2.
TheO’Nan–Scotttheoremdescribesthestructureoffiniteprimitivepermutation
groups:therearefivefamiliesofthese.Thus,toproveConjecture1.1,itissufficient
toproveitforeachofthesefamilies.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
N.Gilletal.,Cherlin’sConjectureforFinitePrimitiveBinaryPermutationGroups,
LectureNotesinMathematics2302,https://doi.org/10.1007/978-3-030-95956-2_1
