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Acknowledgements
Introduction
Overview of Facilities
The Construction of Finitely Presented Groups
Free Groups and Words
Construction of a Free Group
Construction of Words
Access Functions for Words
Arithmetic Operators for Words
Comparison of Words
Relations
Construction of an FP-Group
The Quotient Group Constructor
The FP-Group Constructor
Construction from a Finite Permutation or Matrix Group
Construction of the Standard Presentation for a Coxeter Group
Conversion from a Special Form of FP-Group
Construction of a Standard Group
Construction of Extensions
Accessing the Defining Generators and Relations
Homomorphisms
General Remarks
Construction of Homomorphisms
Accessing Homomorphisms
Computing Homomorphisms to Permutation Groups
Computing Homomorphisms to Permutation Groups Interactively
Finding Homomorphisms onto Simple Groups
Searching for Isomorphisms
Abelian, Nilpotent and Soluble Quotient
Abelian Quotient
p-Quotient
The Construction of a p-Quotient
Nilpotent Quotient
Soluble Quotient
Subgroups
Specification of a Subgroup
Index of a Subgroup: The Todd- Coxeter Algorithm
Implicit Invocation of the Todd- Coxeter Algorithm
Constructing a Presentation for a Subgroup
Introduction
Rewriting
Subgroups of Finite Index
Low Index Subgroups
Subgroup Constructions
Properties of Subgroups
Coset Spaces and Tables
Coset Tables
Coset Spaces: Construction
Coset Spaces: Elementary Operations
Accessing Information
Double Coset Spaces: Construction
Coset Spaces: Selection of Cosets
Coset Spaces: Induced Homomorphism
Simplification
Reducing Generating Sets
Tietze Transformations
Representation Theory
Small Group Identification
Bibliography
DETAILS
Introduction
Overview of Facilities
The Construction of Finitely Presented Groups
Free Groups and Words
Construction of a Free Group
FreeGroup(n) : RngIntElt -> GrpFP
Example GrpFP_1_Free (H61E1)
Construction of Words
G ! [ i1, ..., is ] : GrpFP, [ RngIntElt ] -> GrpFPElt
Identity(G) : GrpFP -> GrpFPElt
Random(G, m, n) : GrpFP, RngIntElt, RngIntElt -> GrpFPElt
Access Functions for Words
# w : GrpFPElt -> RngIntElt
ElementToSequence(w) : GrpFPElt -> [ RngIntElt ]
ExponentSum(w, x) : GrpFPElt, GrpFPElt -> RngIntElt
GeneratorNumber(w) : GrpFPElt -> RngIntElt
LeadingGenerator(w) : GrpFPElt -> GrpFPElt
Parent(w) : GrpFPElt -> GrpFP
Example GrpFP_1_WordAccess (H61E2)
Arithmetic Operators for Words
u * v : GrpFPElt, GrpFPElt -> GrpFPElt
u ^ n : GrpFPElt, RngIntElt -> GrpFPElt
u ^ v : GrpFPElt, GrpFPElt -> GrpFPElt
(u, v) : GrpFPElt, GrpFPElt -> GrpFPElt
(u1, ..., un) : List(GrpFPElt) -> GrpFPElt
Comparison of Words
u eq v : GrpFPElt, GrpFPElt -> BoolElt
u ne v : GrpFPElt, GrpFPElt -> BoolElt
u lt v : GrpFPElt, GrpFPElt -> BoolElt
u le v : GrpFPElt, GrpFPElt -> BoolElt
u ge v : GrpFPElt, GrpFPElt -> BoolElt
u gt v : GrpFPElt, GrpFPElt -> BoolElt
Example GrpFP_1_Words (H61E3)
Relations
w1 = w2 : GrpFPElt, GrpFPElt -> GrpFPRel
LHS(r) : RelElt -> GrpFPElt
RHS(r) : RelElt -> GrpFPElt
r[1] = w : GrpFPRel, RngIntElt, GrpFPElt -> GrpFPRel
r[2] = w : GrpFPRel, RngIntElt, GrpFPElt -> GrpFPRel
f(r) : Hom(GrpFP), GrpFPRel -> GrpFPRel
Parent(r) : RelElt -> GrpFP
Example GrpFP_1_Relations (H61E4)
Construction of an FP-Group
The Quotient Group Constructor
quo< F | R > : GrpFP, List -> GrpFP, Hom(Grp)
G / H : GrpFP, GrpFP -> GrpFP
Example GrpFP_1_Symmetric1 (H61E5)
Example GrpFP_1_Symmetric2 (H61E6)
Example GrpFP_1_Modular (H61E7)
The FP-Group Constructor
Group< X | R > : List(Var), List(GrpFPRel) -> GrpFP, Hom(Grp)
Example GrpFP_1_Tetrahedral (H61E8)
Example GrpFP_1_ThreeInvols (H61E9)
Example GrpFP_1_Coxeter (H61E10)
Construction from a Finite Permutation or Matrix Group
FPGroup(G) : GrpPerm -> GrpFP, Hom(Grp)
FPGroupStrong(G) : GrpPerm -> GrpFP, Hom(Grp)
FPGroupStrong(G, N) : GrpPerm, GrpPerm -> GrpFP, Hom(Grp)
Example GrpFP_1_FPGroup1 (H61E11)
Construction of the Standard Presentation for a Coxeter Group
CoxeterGroup(GrpFP, W) : Cat, GrpFPCox -> GrpFP, Map
Example GrpFP_1_FPCoxeterGroups (H61E12)
Conversion from a Special Form of FP-Group
FPGroup(G) : GrpPC -> GrpFP, Hom(Grp)
Example GrpFP_1_FPGroup2 (H61E13)
Construction of a Standard Group
AbelianGroup(GrpFP, [n1,...,nr]): Cat, [ RngIntElt ] -> GrpFP
AlternatingGroup(GrpFP, n) : Cat, RngIntElt -> GrpFP
BraidGroup(GrpFP, n) : Cat, RngIntElt -> GrpFP
CoxeterGroup(GrpFP, t) : Cat, MonStgElt -> GrpFP
CyclicGroup(GrpFP, n) : Cat, RngIntElt -> GrpFP
DihedralGroup(GrpFP, n) : Cat, RngIntElt -> GrpFP
ExtraSpecialGroup(GrpFP, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpFP
SymmetricGroup(GrpFP, n) : Cat, RngIntElt -> GrpFP
Example GrpFP_1_StandardGroups (H61E14)
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_1_ControlExtn (H61E15)
Example GrpFP_1_DirectProduct (H61E16)
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 ]
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_1_Homomorphism (H61E17)
Computing Homomorphisms to Permutation Groups
Homomorphisms(F, G, A : parameters) : GrpFP, GrpPerm, GrpPerm -> [ HomGrp ]
Example GrpFP_1_Homomorphisms1 (H61E18)
Computing Homomorphisms to Permutation Groups Interactively
HomomorphismsProcess(F, G, A : parameters) : GrpFP, GrpPerm, GrpPerm -> GrpFPHomsProc
NextElement(~P) : GrpFPHomsProc ->
Complete(~P) : GrpFPHomsProc ->
IsEmpty(P) : GrpFPHomsProc -> BoolElt
IsValid(P) : GrpFPHomsProc -> BoolElt
DefinesHomomorphism(P) : GrpFPHomsProc -> BoolElt
Homomorphism(P) : GrpFPHomsProc -> HomGrp
# P : GrpFPHomsProc -> RngIntElt
Homomorphisms(P) : GrpFPHomsProc -> [ HomGrp ]
Example GrpFP_1_Homomorphisms2 (H61E19)
Example GrpFP_1_Homomorphisms2-2 (H61E20)
Finding Homomorphisms onto Simple Groups
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_1_SimpleQuotients (H61E21)
Searching for Isomorphisms
SearchForIsomorphism(F, G, m : parameters) : GrpFP, GrpFP, RngIntElt -> BoolElt, HomGrp, HomGrp
Example GrpFP_1_SearchForIso1 (H61E22)
Example GrpFP_1_SearchForIso2 (H61E23)
Abelian, Nilpotent and Soluble Quotient
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
TorsionFreeRank(G) : GrpFP -> RngIntElt
Example GrpFP_1_F27 (H61E24)
Example GrpFP_1_modular-abelian-quotient (H61E25)
p-Quotient
The Construction of a p-Quotient
pQuotient(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum , BoolElt
Example GrpFP_1_pQuotient1 (H61E26)
Example GrpFP_1_pQuotient2 (H61E27)
Example GrpFP_1_pQuotient3 (H61E28)
Example GrpFP_1_pQuotient4 (H61E29)
Nilpotent Quotient
NilpotentQuotient(G, c: parameters) : GrpFP, RngIntElt -> GrpGPC, Map
Example GrpFP_1_NilpotentQuotient0 (H61E30)
Example GrpFP_1_NilpotentQuotient1 (H61E31)
Example GrpFP_1_NilpotentQuotient2 (H61E32)
SetVerbose("NilpotentQuotient", n) : MonStgElt, RngIntElt ->
Example GrpFP_1_NilpotentQuotient3 (H61E33)
Soluble Quotient
SolvableQuotient(G : parameters): GrpFP, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
Example GrpFP_1_SolubleQuotient1 (H61E34)
SolvableQuotient(F, n : parameters): GrpFP, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
Example GrpFP_1_SolubleQuotient2 (H61E35)
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
DerivedSubgroup(G) : GrpGPC -> GrpGPC
Example GrpFP_1_Subgroups1 (H61E36)
Example GrpFP_1_Subgroups2 (H61E37)
Index of a Subgroup: The Todd- Coxeter Algorithm
ToddCoxeter(G, H: parameters) : GrpFP, GrpFP -> RngIntElt, Map, RngIntElt, RngIntElt
Index(G, H: parameters) : GrpFP, GrpFP -> RngIntElt
Example GrpFP_1_Index1 (H61E38)
Order(G: parameters) : GrpFP -> RngIntElt
Example GrpFP_1_Order11 (H61E39)
Example GrpFP_1_HN (H61E40)
Example GrpFP_1_Family (H61E41)
Implicit Invocation of the Todd- Coxeter Algorithm
SetGlobalTCParameters(: parameters) : ->
UnsetGlobalTCParameters() : ->
Example GrpFP_1_ImplicitCosetEnumeration (H61E42)
Constructing a Presentation for a Subgroup
Introduction
Rewriting
Rewrite(G, H : parameters) : GrpFP, GrpFP -> GrpFP, Map
Rewrite(G, ~H : parameters) : GrpFP, GrpFP ->
Example GrpFP_1_Rewrite (H61E43)
Example GrpFP_1_Rewrite2 (H61E44)
Subgroups of Finite Index
Low Index Subgroups
LowIndexSubgroups(G, R : parameters) : GrpFP, RngIntElt -> [ GrpFP ]
Example GrpFP_1_Lix1 (H61E45)
Example GrpFP_1_Lix2 (H61E46)
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_1_Lix3 (H61E47)
Example GrpFP_1_Lix4 (H61E48)
Example GrpFP_1_Lix5 (H61E49)
LowIndexNormalSubgroups(G, n: parameters) : GrpFP, RngIntElt -> [ Rec ]
Subgroup Constructions
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_1_SubgroupConstructions (H61E50)
Example GrpFP_1_SchreierGenerators (H61E51)
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_1_SubgroupOps (H61E52)
Example GrpFP_1_BuildSubgroups (H61E53)
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_1_CosetTable1 (H61E54)
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_1_CosetTable2 (H61E55)
Example GrpFP_1_CosetSpace (H61E56)
Example GrpFP_1_DerSub (H61E57)
Example GrpFP_1_ExcludedConjugates (H61E58)
Double Coset Spaces: Construction
DoubleCoset(G, H, g, K ) : GrpFP, GrpFP, GrpFPElt, GrpFP -> GrpFPDcosElt
DoubleCosets(G, H, K) : GrpFP, GrpFP, GrpFP -> { GrpFPDcosElt }
Example GrpFP_1_DoubleCosets (H61E59)
Coset Spaces: Selection of Cosets
CosetsSatisfying(T, S: parameters) : Map, { GrpFPElt }: -> { GrpFPCosElt }
Example GrpFP_1_CosetSatisfying (H61E60)
Coset Spaces: Induced Homomorphism
CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, Grp
CosetAction(V) : GrpFPCos, Grp -> Hom(Grp), GrpPerm
CosetImage(G, H) : Grp, Grp -> GrpPerm
CosetImage(V) : GrpFPCos, Grp -> GrpPerm
CosetKernel(G, H) : GrpFP, GrpFP -> GrpFP
CosetKernel(V) : GrpFPCos -> GrpFP
Example GrpFP_1_Co1 (H61E61)
Example GrpFP_1_G23 (H61E62)
Simplification
Reducing Generating Sets
ReduceGenerators(G) : GrpFP -> GrpFP, Map
Tietze Transformations
Simplify(G: parameters) : GrpFP -> GrpFP, Map
Example GrpFP_1_Simplify1 (H61E63)
SimplifyLength(G: parameters) : GrpFP -> GrpFP, Map
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_1_F276 (H61E64)
Example GrpFP_1_ReduceGeneratingSet (H61E65)
Example GrpFP_1_F29 (H61E66)
Example GrpFP_1_L372 (H61E67)
Representation Theory
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_1_RepresentationTheory (H61E68)
Example GrpFP_1_gmoduleprimes (H61E69)
Small Group Identification
IdentifyGroup(G): GrpFP -> Tup
Bibliography
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