- Introduction
- Creation of a Group
- Construction Functions
- CyclicGroup(GrpPC, n) : Cat, RngIntElt -> GrpPC
- AbelianGroup(GrpPC, Q) : Cat, [RngIntElt] -> GrpPC
- DihedralGroup(GrpPC, n) : Cat, RngIntElt -> GrpPC
- ExtraSpecialGroup(GrpPC, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpPC
- Example GrpPC_Standard (H69E1)
- Definition by Presentation
- Possibly Inconsistent Presentations
- Basic Group Properties
- Homomorphisms
- New Groups from Existing
- DirectProduct(G, H) : GrpPC, GrpPC -> GrpPC, [Map], [Map]
- DirectProduct(Q) : [GrpPC] -> GrpPC, [ Map ], [ Map ]
- Extension(G, H, f) : GrpPC, GrpPC, [Map] -> GrpPC
- Extension(M, H) : ModGrp, GrpPC -> GrpPC
- Extension(G, H, f, t) : GrpPC, GrpPC, [Map], [GrpPCElt] -> GrpPC
- Extension(M, H, t) : ModGrp, GrpPC, [ModGrpElt] -> GrpPC
- IsExtension(G, H, f) : GrpPC, GrpPC, [Map] -> BoolElt, GrpPC
- WreathProduct(G, H) : GrpPC, GrpPC -> GrpPC
- WreathProduct(G, H, f) : GrpPC, GrpPC, Map -> GrpPC
- Example GrpPC_extension (H69E6)
- Example GrpPC_cossey_hawkes (H69E7)
- Elements
- Definition of Elements
- Arithmetic Operations on Elements
- g * h : GrpPCElt, GrpPCElt -> GrpPCElt
- g *:= h : GrpPCElt, GrpPCElt -> GrpPCElt
- g ^ n: GrpPCElt, RngIntElt -> GrpPCElt
- g ^:= n: GrpPCElt, RngIntElt -> GrpPCElt
- g / h : GrpPCElt, GrpPCElt -> GrpPCElt
- g /:= h : GrpPCElt, GrpPCElt -> GrpPCElt
- g ^ h : GrpPCElt, GrpPCElt -> GrpPCElt
- g ^:= h : GrpPCElt, GrpPCElt -> GrpPCElt
- (g1, ..., gn) : List(GrpPCElt) -> GrpPCElt
- Properties of Elements
- Predicates for Elements
- Set Operations
- Conjugacy
- Class(H, g) : GrpPC, GrpPCElt -> { GrpPCElt }
- ConjugacyClasses(G) : GrpPC -> [ <RngIntElt, RngIntElt, GrpPCElt> ]
- ClassMap(G) : GrpPC -> Map
- ClassRepresentative(G, x) : GrpPC, GrpPCElt -> GrpPCElt
- ClassCentraliser(G, i) : GrpPC, RngIntElt -> GrpPCElt
- IsConjugate(G, g, h) : GrpPC, GrpPCElt, GrpPCElt -> BoolElt, GrpPCElt
- NumberOfClasses(G) : GrpPC -> RngIntElt
- PowerMap(G) : GrpPC -> Map
- Example GrpPC_class_map (H69E12)
- Subgroups
- Definition of Subgroups by Generators
- Membership and Coercion
- Inclusion and Equality
- Standard Subgroup Constructions
- H ^ g : GrpPC, GrpPCElt -> GrpPC
- H meet K : GrpPC, GrpPC -> GrpPC
- H meet:= K : GrpPC, GrpPC -> GrpPC
- CommutatorSubgroup(G, H, K) : GrpPC, GrpPC, GrpPC -> GrpPC
- Centralizer(G, g) : GrpPC, GrpPCElt -> GrpPC
- Centralizer(G, H) : GrpPC, GrpPC -> GrpPC
- Core(G, H) : GrpPC, GrpPC -> GrpPC
- H ^ G : GrpPC, GrpPC -> GrpPC
- Normalizer(G, H) : GrpPC, GrpPC -> GrpPC
- Example GrpPC_subgroup-constructions (H69E15)
- Properties of Subgroups
- Predicates for Subgroups
- IsCentral(G, H) : GrpPC, GrpPC -> BoolElt
- IsConjugate(G, H, K) : GrpPC, GrpPC, GrpPC -> BoolElt, GrpPCElt
- IsMaximal(G, H) : GrpPC, GrpPC -> BoolElt
- IsNormal(G, H) : GrpPC, GrpPC -> BoolElt
- IsSelfNormalizing(G, H) : GrpPC, GrpPC -> BoolElt
- IsSubnormal(G, H) : GrpPC, GrpPC -> BoolElt
- Example GrpPC_sub-predicates (H69E16)
- Hall π-Subgroups and Sylow Systems
- Conjugacy Classes of Subgroups
- Quotient Groups
- Normal Subgroups and Subgroup Series
- Cosets
- Coset Tables and Transversals
- Transversal(G, H) : GrpPC, GrpPC -> {@ GrpPCElt @}, Map
- CosetTable(G, H) : GrpPC, GrpPC -> Map
- Transversal(G, H, K) : GrpPC, GrpPC, GrpPC -> {@ GrpPCElt @}, Map
- ShortCosets(p, H, G) : GrpPCElt, GrpPC, GrpPC -> [GrpPCElt]
- Action on a Coset Space
- Automorphism Group
- Generating p-groups
- Representation Theory
- CharacterDegrees(G) : GrpPC -> [ Tup ]
- CharacterDegrees(G) : GrpFin -> [ Tup ]
- CharacterDegreesPGroup(G) : GrpFin -> [ RngIntElt ]
- CharacterTable(G: parameters) : GrpPC -> TabChtr
- CharacterTableConlon(G) : GrpPC -> [ AlgChtrElt ]
- GModule(G, M) : GrpPC, AlgMat -> ModAlg
- GModule(G, A) : GrpPC, GrpPC -> ModAlg, Map
- GModule(G, A, B) : GrpPC, GrpPC, GrpPC -> ModAlg, Map
- AbsolutelyIrreducibleRepresentationsSchur(G, k: parameters) : GrpPC, Rng -> List[Map]
- IrreducibleRepresentationsSchur(G, k: parameters) : GrpPC, Rng -> List[Map]
- Example GrpPC_Reps (H69E31)
- Central Extensions
- ExtGenerators(G, U) : GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]
- HomGenerators(G, U) : GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]
- ElementSequence(G) : GrpPC -> SeqEnum
- RepresentativeCocycles(G, U, Ext, Hom) : GrpPC, GrpPC, [AlgMatElt], [AlgMatElt]-> [AlgMatElt]
- CentralExtension(G, U, A) : GrpPC, GrpPC, AlgMatElt -> GrpPC
- CentralExtensions(G, U, Q) : GrpPC, GrpPC, [AlgMatElt] -> [GrpPC]
- CentralExtensionProcess(G, U) : GrpPC, GrpPC -> Proc
- NextExtension(~P) : Rec -> GrpPC
- IsEmpty(P) : Rec -> BoolElt
- Example GrpPC_CentralExtension (H69E32)
- Transfer Between Group Categories
- More About Presentations
- Conditioned Presentations
- Special Presentations
- SpecialPresentation(G) : GrpPC -> GrpPC
- SpecialWeights(G) : GrpPC -> [ <RngIntElt, RngIntElt, RngIntElt> ]
- NilpotentLength(G) : GrpPC -> RngIntElt
- NilpotentBoundary(G,i) : GrpPC, RngIntElt -> RngIntElt
- MinorLength(G,i) : GrpPC, RngIntElt -> RngIntElt
- MinorBoundary(G,i,j) : GrpPC, RngIntElt, RngIntElt -> RngIntElt
- LayerLength(G,i,j) : GrpPC, RngIntElt, RngIntElt -> RngIntElt
- LayerBoundary(G,i,j,k) : GrpPC, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
- Example GrpPC_SpecialPresentation (H69E35)
- CompactPresentation
- Optimizing Magma Code
- p-Groups of Tame Genus
- Verbose Printing
- Constructors
- TGRandomGroup(q, n, g : parameters) : RngIntElt, RngIntElt, RngIntElt -> GrpPC
- Example GrpPC_RandomGenusGroups (H69E39)
- RandomGenus2Group(q, d : parameters) : RngIntElt, [RngIntElt] -> GrpPC
- Example GrpPC_PrescribedBlocks (H69E40)
- RandomGenus1Group(q, d, r : parameters) : RngIntElt, RngIntElt, RngIntElt -> GrpPC
- Example GrpPC_Heisenbergs (H69E41)
- Genus2Group(f) : RngUPolElt -> GrpPC
- Example GrpPC_Pfaffians (H69E42)
- Direct Indecomposability
- Genus
- Isomorphism
- Automorphism Groups
- Canonical Labelling
- Bibliography
V2.28, 13 July 2023