FINITELY PRESENTED GROUPS
Acknowledgements
Introduction
Overview of Facilities
Construction of an FP-Group
Introduction
Quotient Group Constructor
The FP-Group Constructor
Accessing the Defining Generators and Relations
Operations on Words
Operations on Presentations
Simplification
Automatic Simplification
Interactive Simplification
Standard Constructions
Familiar Groups as FP-Groups
Construction of Extensions
Conversion to FP-Groups
Properties of an FP-Group
Cardinality
Small Cancellation Conditions
Largeness
Subgroups
Specification of a Subgroup
The Todd-Coxeter Algorithm
Interactive Coset Enumeration
Introduction
Constructing and Modifying a Coset Enumeration Process
Starting and Restarting an Enumeration
Accessing Information
Implicit Invocation of the Todd- Coxeter Algorithm
Coset Spaces and Tables
Coset Tables
Coset Spaces: Induced Homomorphism
Coset Spaces: Construction
Coset Spaces: Elementary Operations
Accessing Information
Double Coset Spaces: Construction
Coset Spaces: Selection of Cosets
Constructing a Presentation for a Subgroup
Subgroups of Finite Index
Low Index Subgroups
Operations for Subgroups of Finite Index
Properties of Subgroups
Finite FP-Groups
Concrete Representations
The Cayley Graph
Homomorphisms
General Remarks
Construction of Homomorphisms
Accessing Homomorphisms
Constructing Homomorphisms onto Finite Groups
Searching for Isomorphisms
Quotient Group Methods
Abelian Quotient
p-Quotient
p-Quotient Process
Using p-Quotient Interactively
Nilpotent Quotient
Soluble Quotient
Soluble Quotient Advanced
Introduction
Construction
Calculating the Relevant Primes
The Intrinsics
Simple Group Quotients
The (L)2-Quotient Algorithm
Basic Usage
Intermediate Usage
Advanced Usage
Handling Infinite (L)2-quotients
Infinite L2 Quotients
The (L)3(U)3-Quotient Algorithm
KG-Modules
Some Developed Examples
Bibliography
Introduction
Overview of Facilities
Construction of an FP-Group
Introduction
Quotient Group Constructor
quo< F | R > : GrpFP, List -> GrpFP, Hom(Grp)
G / H : GrpFP, GrpFP -> GrpFP
Example GrpFP_Symmetric1 (H80E1)
Example GrpFP_Symmetric2 (H80E2)
Example GrpFP_Modular (H80E3)
The FP-Group Constructor
FPGroup< X | R > : List(Var), List(GrpFPRel) -> GrpFP, Hom(Grp)
Example GrpFP_Tetrahedral (H80E4)
Example GrpFP_ThreeInvols (H80E5)
Example GrpFP_Coxeter (H80E6)
Accessing the Defining Generators and Relations
G . i : GrpFP, RngIntElt -> GrpFPElt
Generators(G) : GrpFP -> { GrpFPElt }
NumberOfGenerators(G) : GrpFP -> RngIntElt
PresentationLength(G) : GrpFP -> RngIntElt
Relations(G) : GrpFP -> [ GrpFPRel ]
Operations on Words
Eliminate(u, x, v) : GrpFPElt, GrpFPElt, GrpFPElt -> GrpFPElt
Eliminate(U, x, v) : { GrpFPElt }, GrpFPElt, GrpFPElt -> { GrpFPElt }
Match(u, v, f) : GrpFPElt, GrpFPElt, RngIntElt -> BoolElt, RngIntElt
RotateWord(u, n) : GrpFPElt, RngIntElt -> GrpFPElt
Substitute(u, f, n, v) : GrpFPElt, RngIntElt, RngIntElt, GrpFPElt -> GrpFPElt
Subword(u, f, n) : GrpFPElt, RngIntElt, RngIntElt -> GrpFPElt
Example GrpFP_WordOps (H80E7)
Operations on Presentations
AddGenerator(G) : GrpFP -> GrpFP
AddGenerator(G, w) : GrpFP, GrpFPElt -> GrpFP
AddRelation(G, r) : GrpFP, RelElt -> GrpFP
AddRelation(G, g) : GrpFP, GrpFPElt -> GrpFP
AddRelation(G, r, i) : GrpFP, RelElt, RngIntElt -> GrpFP
AddRelation(G, g, i) : GrpFP, GrpFPElt, RngIntElt -> GrpFP
DeleteGenerator(G, x) : GrpFP, GrpFPElt -> GrpFP
DeleteRelation(G, r) : GrpFP, RelElt -> GrpFP
DeleteRelation(G, g) : GrpFP, GrpFPElt -> GrpFP
DeleteRelation(G, i) : GrpFP, RngIntElt -> GrpFP
ReplaceRelation(G, s, r) : GrpFP, RelElt, RelElt -> GrpFP
ReplaceRelation(G, i, r) : GrpFP, RngIntElt, RelElt -> GrpFP
ReplaceRelation(G, i, g) : GrpFP, RngIntElt, GrpFPElt -> GrpFP
Example GrpFP_Replace (H80E8)
Simplification
Automatic Simplification
ReduceGenerators(G) : GrpFP -> GrpFP, Map
Simplify(G: parameters) : GrpFP -> GrpFP, Map
Example GrpFP_Simplify1 (H80E9)
SimplifyLength(G: parameters) : GrpFP -> GrpFP, Map
Interactive Simplification
TietzeProcess(G: parameters) : GrpFP -> Process(Tietze)
ShowOptions(~P : parameters) : GrpFPTietzeProc ->
SetOptions(~P : parameters) : GrpFPTietzeProc ->
Simplify(~P : parameters) : GrpFPTietzeProc ->
SimplifyLength(~P : parameters) : GrpFPTietzeProc ->
Eliminate(~P: parameters) : GrpFPTietzeProc ->
Search(~P: parameters) : GrpFPTietzeProc ->
SearchEqual(~P: parameters) : GrpFPTietzeProc ->
Group(P) : GrpFPTietzeProc -> GrpFP, Map
NumberOfGenerators(P) : GrpFPTietzeProc -> RngIntElt
NumberOfRelations(P) : GrpFPTietzeProc -> RngIntElt
PresentationLength(P) : GrpFPTietzeProc -> RngIntElt
Example GrpFP_F276 (H80E10)
Example GrpFP_ReduceGeneratingSet (H80E11)
Standard Constructions
Familiar Groups as FP-Groups
AbelianFPGroup([n1,...,nr]): [ RngIntElt ] -> GrpFP
AlternatingFPGroup(n) : RngIntElt -> GrpFP
BraidFPGroup(n) : RngIntElt -> GrpFP
CoxeterFPGroup(t) : MonStgElt -> GrpFP
CyclicFPGroup(n) : RngIntElt -> GrpFP
DihedralFPGroup(n) : RngIntElt -> GrpFP
ExtraSpecialFPGroup(p, n : parameters) : RngIntElt, RngIntElt -> GrpFP
SymmetricFPGroup(n) : RngIntElt -> GrpFP
Example GrpFP_StandardGroups (H80E12)
Construction of Extensions
Darstellungsgruppe(G) : GrpFP -> GrpFP
DirectProduct(G, H) : GrpFP, GrpFP -> GrpFP
DirectProduct(Q) : [ GrpFP ] -> GrpFP
FreeProduct(G, H) : GrpFP, GrpFP -> GrpFP
FreeProduct(Q) : [ GrpFP ] -> GrpFP
Example GrpFP_ControlExtn (H80E13)
Example GrpFP_DirectProduct (H80E14)
Conversion to FP-Groups
FPGroup(G) : GrpPerm -> GrpFP, Hom(Grp)
Example GrpFP_FPGroup1 (H80E15)
FPGroupStrong(G) : GrpPerm -> GrpFP, Hom(Grp)
FPGroupStrong(G, N) : GrpPerm, GrpPerm -> GrpFP, Hom(Grp)
Example GrpFP_FPGroup2 (H80E16)
FPGroup(G) : GrpPC -> GrpFP, Hom(Grp)
Example GrpFP_FPGroup2 (H80E17)
CoxeterFPGroup(W) : GrpFPCox -> GrpFP, Map
Example GrpFP_FPCoxeterGroups (H80E18)
Properties of an FP-Group
Cardinality
Order(G: parameters) : GrpFP -> RngIntElt
Example GrpFP_Order11 (H80E19)
IsInfiniteFPGroup(G : parameters) : GrpFP -> BoolElt
Example GrpFP_ProveInf1 (H80E20)
HasPositiveH1Dimension(G, phi : parameters) : GrpFP, HomGrp -> BoolElt
Small Cancellation Conditions
SmallCancellationConditions(G) : GrpFP -> RngIntElt, RnIntElt,FldRatElt)
Example GrpFP_small_cancel (H80E21)
Largeness
IsLarge(G, L, U:parameters) : GrpFP, RngIntElt, RngIntElt -> BoolElt, GrpFP
Subgroups
Specification of a Subgroup
sub< G | L > : GrpFP, List -> GrpFP
sub< G | f > : GrpFP, Hom(Grp) -> GrpFP
ncl< G | L > : GrpFP, List -> GrpFP
ncl<G | f> : GrpFP, Hom(Grp) -> GrpFP
CommutatorSubgroup(G) : GrpFP -> GrpFP
Example GrpFP_Subgroups1 (H80E22)
Example GrpFP_Subgroups2 (H80E23)
The Todd-Coxeter Algorithm
ToddCoxeter(G, H: parameters) : GrpFP, GrpFP -> RngIntElt, Map, RngIntElt, RngIntElt
Index(G, H: parameters) : GrpFP, GrpFP -> RngIntElt
Example GrpFP_Index1 (H80E24)
Example GrpFP_HN (H80E25)
Example GrpFP_Family (H80E26)
Interactive Coset Enumeration
Introduction
Constructing and Modifying a Coset Enumeration Process
CosetEnumerationProcess(G, H: parameters) : GrpFP, GrpFP -> GrpFPCosetEnumProc
AddRelator(~P, w) : GrpFPCosetEnumProc, GrpFPElt ->
AddSubgroupGenerator(~P, w) : GrpFPCosetEnumProc, GrpFPElt ->
SetProcessParameters(~P: parameters) : GrpFPCosetEnumProc ->
Starting and Restarting an Enumeration
StartEnumeration(~P: parameters) : GrpFPCosetEnumProc ->
RedoEnumeration(~P: parameters) : GrpFPCosetEnumProc ->
CanRedoEnumeration(P) : GrpFPCosetEnumProc -> BoolElt
ContinueEnumeration(~P: parameters) : GrpFPCosetEnumProc ->
CanContinueEnumeration(P) : GrpFPCosetEnumProc -> BoolElt
ResumeEnumeration(~P: parameters) : GrpFPCosetEnumProc ->
Accessing Information
CosetsSatisfying(P : parameters) : GrpFPCosetEnumProc -> { GrpFPElt }
CosetTable(P) : GrpFPCosetEnumProc -> Map
HasValidCosetTable(P) : GrpFPCosetEnumProc -> BoolElt
HasClosedCosetTable(P) : GrpFPCosetEnumProc -> BoolElt
ExcludedConjugate(P) : GrpFPCosetEnumProc -> GrpFPElt
ExistsCosetSatisfying(P : parameters) : GrpFPCosetEnumProc -> BoolElt, GrpFPElt
ExistsExcludedConjugate(P) : GrpFPCosetEnumProc -> BoolElt, GrpFPElt
ExistsNormalisingCoset(P) : GrpFPCosetEnumProc -> BoolElt, GrpFPElt
Group(P) : GrpFPCosetEnumProc -> GrpFP
Index(P) : GrpFPCosetEnumProc -> RngIntElt
HasValidIndex(P) : GrpFPCosetEnumProc -> BoolElt
MaximalNumberOfCosets(P) : GrpFPCosetEnumProc -> RngIntElt
Subgroup(P) : GrpFPCosetEnumProc -> GrpFP
TotalNumberOfCosets(P) : GrpFPCosetEnumProc -> RngIntElt
Example GrpFP_ACEProc1 (H80E27)
Example GrpFP_ACEProc2 (H80E28)
Example GrpFP_ACEProc3 (H80E29)
Example GrpFP_ACEProc4 (H80E30)
Implicit Invocation of the Todd- Coxeter Algorithm
SetGlobalTCParameters(: parameters) : ->
UnsetGlobalTCParameters() : ->
Example GrpFP_ImplicitCosetEnumeration (H80E31)
Coset Spaces and Tables
Coset Tables
CosetTable(G, H: parameters) : GrpFP, GrpFP -> Map
CosetTableToRepresentation(G, T): GrpFP, Map -> Map, GrpPerm, Grp
CosetTableToPermutationGroup(G, T) : GrpFP, Map -> GrpPerm
Example GrpFP_CosetTable1 (H80E32)
Coset Spaces: Induced Homomorphism
CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, Grp
CosetAction(V) : GrpFPCos -> Hom(Grp), GrpPerm
CosetImage(G, H) : Grp, Grp -> GrpPerm
CosetImage(V) : GrpFPCos -> GrpPerm
CosetKernel(G, H) : GrpFP, GrpFP -> GrpFP
CosetKernel(V) : GrpFPCos -> GrpFP
Example GrpFP_Co1 (H80E33)
Example GrpFP_G23 (H80E34)
Coset Spaces: Construction
CosetSpace(G, H: parameters) : GrpFP, GrpFP: -> GrpFPCos
RightCosetSpace(G, H: parameters) : GrpFP, GrpFP -> GrpFPCos
Coset Spaces: Elementary Operations
H * g : GrpFP, GrpFPElt -> GrpFPCosElt
C * g : GrpFPCosElt, GrpFPElt -> GrpFPCosElt
C * D : GrpFPCosElt, GrpFPCosElt -> GrpFPCosElt
g in C : GrpFPElt, GrpFPCosElt -> BoolElt
g notin C : GrpFPElt, GrpFPCosElt -> BoolElt
C1 eq C2 : GrpFPCosElt, GrpFPCosElt -> BoolElt
C1 ne C2 : GrpFPCosElt, GrpFPCosElt -> BoolElt
Accessing Information
# V : GrpFPCos -> RngIntElt
Action(V) : GrpFPCos -> Map
<i, w> @ T : GrpFPCosElt, GrpFPElt, Map -> GrpFPElt
ExplicitCoset(V, i) : GrpFPCos, RngIntElt -> GrpFPCosElt
IndexedCoset(V, w) : GrpFPCos, GrpFPElt -> GrpFPCosElt
IndexedCoset(V, C) : GrpFPCos, GrpFPCosElt -> GrpFPCosElt
Group(V) : GrpFPCos -> GrpFP
Subgroup(V) : GrpFPCos -> GrpFP
IsComplete(V) : GrpFPCos -> BoolElt
ExcludedConjugates(V) : GrpFPCos -> { GrpFPElt }
Transversal(G, H) : GrpFP, GrpFP -> {@ GrpFPElt @}, Map
Example GrpFP_CosetTable2 (H80E35)
Example GrpFP_CosetSpace (H80E36)
Example GrpFP_DerSub (H80E37)
Example GrpFP_ExcludedConjugates (H80E38)
Double Coset Spaces: Construction
DoubleCoset(G, H, g, K ) : GrpFP, GrpFP, GrpFPElt, GrpFP -> GrpFPDcosElt
DoubleCosets(G, H, K) : GrpFP, GrpFP, GrpFP -> { GrpFPDcosElt }
Example GrpFP_DoubleCosets (H80E39)
Coset Spaces: Selection of Cosets
CosetsSatisfying(T, S: parameters) : Map, { GrpFPElt }: -> { GrpFPCosElt }
Example GrpFP_CosetSatisfying (H80E40)
Constructing a Presentation for a Subgroup
Rewrite(G, H : parameters) : GrpFP, GrpFP -> GrpFP, Map
Rewrite(G, ~H : parameters) : GrpFP, GrpFP ->
Example GrpFP_Rewrite (H80E41)
Example GrpFP_Rewrite2 (H80E42)
Subgroups of Finite Index
Low Index Subgroups
LowIndexSubgroups(G, R : parameters) : GrpFP, RngIntElt -> [ GrpFP ]
Example GrpFP_Lix1 (H80E43)
Example GrpFP_Lix2 (H80E44)
LowIndexProcess(G, R : parameters) : GrpFP, RngIntElt -> Process(Lix)
NextSubgroup(~P) : GrpFPLixProc ->
ExtractGroup(P) : GrpFPLixProc -> GrpFP
ExtractGenerators(P) : GrpFPLixProc -> { GrpFPElt }
IsEmpty(P) : GrpFPLixProc -> BoolElt
IsValid(P) : GrpFPLixProc -> BoolElt
Example GrpFP_Lix3 (H80E45)
Example GrpFP_Lix4 (H80E46)
Example GrpFP_Lix5 (H80E47)
LowIndexNormalSubgroups(G, n: parameters) : GrpFP, RngIntElt -> [ Rec ]
Operations for Subgroups of Finite Index
H ^ u : GrpFP, GrpFPElt -> GrpFP
H meet K : GrpFP, GrpFP -> GrpFP
Core(G, H) : GrpFP, GrpFP -> GrpFP
GeneratingWords(G, H) : GrpFP, GrpFP -> { GrpFPElt }
MaximalOvergroup(G, H) : GrpFP, GrpFP -> GrpFP
MinimalOvergroup(G, H) : GrpFP, GrpFP -> GrpFP
H ^ G : GrpFP, GrpFP -> GrpFP
Normaliser(G, H) : GrpFP, GrpFP -> GrpFP
SchreierGenerators(G, H : parameters) : GrpFP, GrpFP -> { GrpFPElt }
SchreierSystem(G, H) : GrpFP, GrpFP -> {@ GrpFPElt @}, Map
Transversal(G, H, K) : GrpFP, GrpFP, GrpFP -> {@ GrpFPElt @}, Map
Example GrpFP_SubgroupConstructions (H80E48)
Example GrpFP_SchreierGenerators (H80E49)
Properties of Subgroups
u ∈H : GrpFPElt, GrpFP -> BoolElt
u ∉H : GrpFPElt, GrpFP -> BoolElt
H eq K : GrpFP, GrpFP -> BoolElt
H ≠K : GrpFP, GrpFP -> BoolElt
H ⊂K : GrpFP, GrpFP -> BoolElt
H notsubset K : GrpFP, GrpFP -> BoolElt
IsConjugate(G, H, K) : GrpFP, GrpFP, GrpFP -> BoolElt, GrpFPElt
IsNormal(G, H) : GrpFP, GrpFP -> BoolElt
IsMaximal(G, H) : GrpFP, GrpFP -> BoolElt
IsSelfNormalizing(G, H) : GrpFP, GrpFP -> BoolElt
Example GrpFP_SubgroupOps (H80E50)
Example GrpFP_BuildSubgroups (H80E51)
Finite FP-Groups
Concrete Representations
PermutationGroup(G) : GrpFP -> GrpPerm, GrpHom
PCGroup(G) : GrpFP -> GrpPC, GrpHom
The Cayley Graph
CayleyGraph(G) : GrpFP -> GrphVertSet, GrphEdgeSeto
SchreierGraph(G, H) : Grp, Grp -> Grph, GrphVertSet, GrphEdgeSet
Example GrpFP_cayley-graph (H80E52)
Homomorphisms
General Remarks
Construction of Homomorphisms
hom< P -> G | S > : Struct , Struct -> Map
IsSatisfied(U, E) : { RelElt }, [ GrpElt ] -> BoolElt
Accessing Homomorphisms
w @ f : GrpFPElt, Map -> GrpElt
H @ f : GrpFP, Map -> Grp
g @@ f : GrpElt, Map -> GrpFPElt
H @@ f : Grp, Map -> GrpFP
Domain(f) : Map -> Grp
Codomain(f) : Map -> Grp
Image(f) : Map -> Grp
Kernel(f) : Map -> Grp
Example GrpFP_Homomorphism (H80E53)
Constructing Homomorphisms onto Finite Groups
Homomorphisms(F, G, A : parameters) : GrpFP, GrpPerm, GrpPerm -> [ HomGrp ]
Example GrpFP_Homomorphisms1 (H80E54)
Homomorphisms(F, G, A : parameters) : GrpFP, GrpPC, GrpPC -> [ HomGrp ]
HomomorphismsProcess(F, G, A : parameters) : GrpFP, GrpPerm, GrpPerm -> GrpFPHomsProc
NextElement(~P) : GrpFPHomsProc ->
IsEmpty(P) : GrpFPHomsProc -> BoolElt
IsValid(P) : GrpFPHomsProc -> BoolElt
DefinesHomomorphism(P) : GrpFPHomsProc -> BoolElt
Homomorphism(P) : GrpFPHomsProc -> HomGrp
# P : GrpFPHomsProc -> RngIntElt
Example GrpFP_Homomorphisms2 (H80E55)
Example GrpFP_Homomorphisms2-2 (H80E56)
Searching for Isomorphisms
SearchForIsomorphism(F, G, m : parameters) : GrpFP, GrpFP, RngIntElt -> BoolElt, HomGrp, HomGrp
Example GrpFP_SearchForIso1 (H80E57)
Example GrpFP_SearchForIso2 (H80E58)
Quotient Group Methods
Abelian Quotient
AbelianQuotient(G) : GrpFP -> GrpAb, Map
ElementaryAbelianQuotient(G, p) : GrpFP, RngIntElt -> GrpAb, Map
AbelianQuotientInvariants(G) : GrpFP -> [ RngIntElt ]
AbelianQuotientInvariants(H) : GrpFP -> [ RngIntElt ]
AbelianQuotientInvariants(G, n) : GrpFP, RngIntElt -> [ RngIntElt ]
AbelianQuotientInvariants(H, n) : GrpFP, RngIntElt -> [ RngIntElt ]
HasComputableAbelianQuotient(G) : GrpFP -> BoolElt, GrpAb, Map
HasInfiniteComputableAbelianQuotient(G) : GrpFP -> BoolElt, GrpAb, Map
IsPerfect(G) : GrpFP -> BoolElt
TorsionFreeRank(G) : GrpFP -> RngIntElt
Example GrpFP_F27 (H80E59)
Example GrpFP_modular-abelian-quotient (H80E60)
HasFiniteAbelianQuotient(G) : GrpFP -> [ RngIntElt ]
AQPrimes(G) : GrpFP -> [ RngIntElt ]
p-Quotient
pQuotient(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum , BoolElt
Example GrpFP_pQuotient1 (H80E61)
Example GrpFP_pQuotient2 (H80E62)
Example GrpFP_pQuotient3 (H80E63)
Example GrpFP_pQuotient4 (H80E64)
HaspQuotientDefinitions(G) : GrpPC -> BoolElt
pQuotientDefinitions(G) : GrpPC -> SeqEnum
p-Quotient Process
pQuotientProcess(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> Process
NextClass(~P : parameters) : GrpPCpQuotientProc ->
Using p-Quotient Interactively
StartNewClass(~P: parameters) : GrpPCpQuotientProc ->
Tails(~P: parameters) : GrpPCpQuotientProc ->
Consistency(~P: parameters) : GrpPCpQuotientProc ->
CollectRelations(~P) : GrpPCpQuotientProc ->
ExponentLaw(~P : parameters) : GrpPCpQuotientProc ->
EliminateRedundancy(~P) : GrpPCpQuotientProc ->
Display(P) : GrpPCpQuotientProc ->
RevertClass(~P) : GrpPCpQuotientProc ->
pCoveringGroup(~P) : GrpPCpQuotientProc ->
GeneratorStructure(P) : GrpPCpQuotientProc ->
Jacobi(~P, c, b, a, ~r) : GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt -> RngIntElt ->
Collect(P, Q) : GrpPCpQuotientProc, [ <RngIntElt, RngIntElt> ] -> [ RngIntElt ] ->
EcheloniseWord(~P, ~r) : GrpPCpQuotientProc -> RngIntElt
SetDisplayLevel(~P, Level) : GrpPCpQuotientProc, RngIntElt ->
ExtractGroup(P) : GrpPCpQuotientProc -> GrpPC
Order(P) : GrpPCpQuotientProc -> RngIntElt
FactoredOrder(P) : GrpPCpQuotientProc -> [ <RngIntElt, RngIntElt> ]
NumberOfPCGenerators(P) : GrpPCpQuotientProc -> RngIntElt
pClass(P) : GrpPCpQuotientProc -> RngIntElt
NuclearRank(G) : GrpPC -> RngIntElt
pMultiplicatorRank(G) : GrpPC -> RngIntElt
Example GrpFP_pQuotient5 (H80E65)
Example GrpFP_pQuotient6 (H80E66)
Example GrpFP_pQuotient7 (H80E67)
Example GrpFP_pQuotient8 (H80E68)
Nilpotent Quotient
NilpotentQuotient(G, c: parameters) : GrpFP, RngIntElt -> GrpGPC, Map
Example GrpFP_NilpotentQuotient0 (H80E69)
Example GrpFP_NilpotentQuotient1 (H80E70)
Example GrpFP_NilpotentQuotient2 (H80E71)
SetVerbose("NilpotentQuotient", n) : MonStgElt, RngIntElt ->
Example GrpFP_NilpotentQuotient3 (H80E72)
Soluble Quotient
SolvableQuotient(G : parameters): GrpFP, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
Example GrpFP_SolubleQuotient1 (H80E73)
SolvableQuotient(F, n : parameters): GrpFP, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
Example GrpFP_SolubleQuotient2 (H80E74)
Soluble Quotient Advanced
Introduction
Construction
Calculating the Relevant Primes
The Intrinsics
SolubleQuotient(F, n : parameters): GrpFP, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
Simple Group Quotients
SimpleQuotients(F, deg1, deg2, ord1, ord2: parameters) : GrpFP, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> List
SimpleQuotientProcess(F, deg1, deg2, ord1, ord2: parameters) : GrpFP, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Rec
NextSimpleQuotient(~P) : Rec ->
IsEmptySimpleQuotientProcess(P) : Rec -> BoolElt
SimpleEpimorphisms(P) : Rec -> SeqEnum, Tup
Example GrpFP_SimpleQuotients (H80E75)
The (L)2-Quotient Algorithm
Basic Usage
L2Quotients(G) : GrpFP -> [ L2Quotient ]
Example GrpFP_L2Quotient (H80E76)
GetMatrices(Q) : L2Quotient -> GrpMat, SeqEnum
Example GrpFP_L2QuotientGetMatrices (H80E77)
Example GrpFP_ModularFinite (H80E78)
Example GrpFP_CoxeterFinite (H80E79)
Intermediate Usage
L2Quotients(M) : AlgMatElt -> [ L2Quotient ]
Example GrpFP_L2QuotientCoxeter (H80E80)
L2Quotients(G) : GrpFP -> [ L2Quotient ]
Example GrpFP_L2QuotientexactOrders (H80E81)
Advanced Usage
L2Quotients(G) : GrpFP -> [ L2Quotient ]
Handling Infinite (L)2-quotients
SpecifyCharacteristic(Q, n) : L2Quotient, RngIntElt -> [ L2Quotient ]
Example GrpFP_L2QuotientSpecifyCharacteristic (H80E82)
AddGroupRelations(Q, R) : L2Quotient, [ GrpFPElt ] -> [ L2Quotient ]
Example GrpFP_L2QuotientAddGroupRelations (H80E83)
Example GrpFP_L2QuotientSpecifyAddGroupRelations2 (H80E84)
AddRingRelations(Q, R) : L2Quotient, [ RngMPolElt ] -> [ L2Quotient ]
Example GrpFP_L2QuotientSpecifyAddRingRelations (H80E85)
Infinite L2 Quotients
HasInfinitePSL2Quotient(G) :: GrpFP -> BoolElt, SeqEnum
Example GrpFP_fp-gps:inf-psl2-quot (H80E86)
The (L)3(U)3-Quotient Algorithm
L3Quotients(G: parameters) : GrpFP -> [ L3Quotient ]
SpecifyCharacteristic(Q, p) : L2Quotient, RngIntElt -> [ L2Quotient ]
AddGroupRelations(Q, r) : L2Quotient, [ GrpFPElt ] -> [ L2Quotient ]
GetMatrices(Q) : L2Quotient -> GrpMat
Example GrpFP_L3Quotient (H80E87)
Example GrpFP_L3infinite (H80E88)
KG-Modules
GModulePrimes(G, A) : GrpFP, GrpFP -> SetMulti
GModulePrimes(G, A, B) : GrpFP, GrpFP, GrpFP -> SetMulti
GModule(G, A, p) : GrpFP, GrpFP, RngIntElt -> ModGrp, Map
GModule(G, A, B, p) : GrpFP, GrpFP, GrpFP, RngIntElt -> ModGrp, Map
Pullback(f, N) : Map, ModGrp -> GrpFP
Example GrpFP_RepresentationTheory (H80E89)
Example GrpFP_gmoduleprimes (H80E90)
Some Developed Examples
Example GrpFP_F29 (H80E91)
Example GrpFP_L372 (H80E92)
Bibliography
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