- Introduction
- Construction of an FP-Group
- Introduction
- Quotient Group Constructor
- The FP-Group Constructor
- Accessing the Defining Generators and Relations
- 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 (H78E7)
- 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 (H78E8)
- Simplification
- Standard Constructions
- Properties of an FP-Group
- Subgroups
- Coset Spaces and Tables
- Coset Tables
- Coset Spaces: Induced Homomorphism
- Coset Spaces: Construction
- Coset Spaces: Elementary Operations
- 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 (H78E35)
- Example GrpFP_CosetSpace (H78E36)
- Example GrpFP_DerSub (H78E37)
- Example GrpFP_ExcludedConjugates (H78E38)
- 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 (H78E39)
- Coset Spaces: Selection of Cosets
- Constructing a Presentation for a Subgroup
- Subgroups of Finite Index
- Low Index Subgroups
- 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 (H78E48)
- Example GrpFP_SchreierGenerators (H78E49)
- 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 (H78E50)
- Example GrpFP_BuildSubgroups (H78E51)
- Finite FP-Groups
- Homomorphisms
- General Remarks
- Construction of Homomorphisms
- Accessing Homomorphisms
- Constructing Homomorphisms onto Finite Groups
- Homomorphisms(F, G, A : parameters) : GrpFP, GrpPerm, GrpPerm -> [ HomGrp ]
- Example GrpFP_Homomorphisms1 (H78E54)
- 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 (H78E55)
- Example GrpFP_Homomorphisms2-2 (H78E56)
- Searching for Isomorphisms
- 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 (H78E59)
- Example GrpFP_modular-abelian-quotient (H78E60)
- HasFiniteAbelianQuotient(G) : GrpFP -> [ RngIntElt ]
- AQPrimes(G) : GrpFP -> [ RngIntElt ]
- p-Quotient
- 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 (H78E65)
- Example GrpFP_pQuotient6 (H78E66)
- Example GrpFP_pQuotient7 (H78E67)
- Example GrpFP_pQuotient8 (H78E68)
- Nilpotent Quotient
- Soluble Quotient
- Soluble Quotient Advanced
- 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 (H78E75)
- The (L)2-Quotient Algorithm
- Infinite L2 Quotients
- The (L)3(U)3-Quotient Algorithm
- 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 (H78E89)
- Example GrpFP_gmoduleprimes (H78E90)
- Some Developed Examples
- Bibliography
V2.28, 13 July 2023