Finite Soluble Groups
- Introduction
- Creation of a Group
- Construction Functions
CyclicGroup(GrpPC, n): Cat, RngIntElt → GrpPCAbelianGroup(GrpPC, Q): Cat, [RngIntElt] → GrpPCDihedralGroup(GrpPC, n): Cat, RngIntElt → GrpPCExtraSpecialGroup(GrpPC, p, n : parameters): Cat, RngIntElt, RngIntElt → GrpPC- Example: Pc-Representation of a Cyclic Group
- Definition by Presentation
- Possibly Inconsistent Presentations
- Basic Group Properties
- Homomorphisms
hom< G -> H | L >: GrpPC, GrpPC, List → MapIsHomomorphism(G, H, L): GrpPC, GrpPC, SeqEnum → BoolElt, MapIsHomomorphism(G, H, L): GrpPC, GrpPC, SetEnum[Tup] → BoolElt, MapIdentityHomomorphism(G): GrpPC → MapKernel(f): Map → GrpPCHomomorphisms(G, H): GrpPC, GrpPC → SeqEnum- Example: Homomorphisms Between Pc-Groups
- 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] → GrpPCExtension(M, H): ModGrp, GrpPC → GrpPCExtension(G, H, f, t): GrpPC, GrpPC, [Map], [GrpPCElt] → GrpPCExtension(G, H, f, t): GrpPC, GrpPC, [Map], {Tup} → GrpPCExtension(M, H, t): ModGrp, GrpPC, [ModGrpElt] → GrpPCIsExtension(G, H, f): GrpPC, GrpPC, [Map] → BoolElt, GrpPCIsExtension(M, H): ModGrp, GrpPC → BoolElt, GrpPCIsExtension(G, H, f, t): GrpPC, GrpPC, [Map], [GrpPCElt] → BoolElt, GrpPCIsExtension(G, H, f, t): GrpPC, GrpPC, [Map], {Tup} → BoolElt, GrpPCIsExtension(M, H, t): ModGrp, GrpPC, [ModGrpElt] → BoolElt, GrpPCWreathProduct(G, H): GrpPC, GrpPC → GrpPCWreathProduct(G, H, f): GrpPC, GrpPC, Map → GrpPCWreathProduct(G, H, f): GrpPC, GrpPC, [GrpPermElt] → GrpPC- Example: Extensions of an Elementary Abelian Group
- Example: Cossey-Hawkes Conjugacy Class Sizes
- Elements
- Definition of Elements
- Arithmetic Operations on Elements
g * h: GrpPCElt, GrpPCElt → GrpPCEltg *:= h: GrpPCElt, GrpPCElt → GrpPCEltg ^ n: GrpPCElt, RngIntElt → GrpPCEltg ^:= n: GrpPCElt, RngIntElt → GrpPCEltg / h: GrpPCElt, GrpPCElt → GrpPCEltg /:= h: GrpPCElt, GrpPCElt → GrpPCEltg ^ h: GrpPCElt, GrpPCElt → GrpPCEltg ^:= h: GrpPCElt, GrpPCElt → GrpPCElt(g₁, ..., gₙ): List(GrpPCElt) → GrpPCElt
- Properties of Elements
- Predicates for Elements
- Set Operations
- Conjugacy
Class(H, g): GrpPC, GrpPCElt → { GrpPCElt}Conjugates(H, g): GrpPC, GrpPCElt → { GrpPCElt}g ^ H: GrpPCElt, GrpPC → { GrpPCElt}ConjugacyClasses(G): GrpPC → [ <RngIntElt, RngIntElt, GrpPCElt> ]Classes(G): GrpPC → [ <RngIntElt, RngIntElt, GrpPCElt> ]ClassMap(G): GrpPC → MapClassRepresentative(G, x): GrpPC, GrpPCElt → GrpPCEltClassRepresentative(G, i): GrpPC, RngIntElt → GrpPCEltClassCentraliser(G, i): GrpPC, RngIntElt → GrpPCEltClassCentralizer(G, i): GrpPC, RngIntElt → GrpPCEltIsConjugate(G, g, h): GrpPC, GrpPCElt, GrpPCElt → BoolElt, GrpPCEltNumberOfClasses(G): GrpPC → RngIntEltNclasses(G): GrpPC → RngIntEltPowerMap(G): GrpPC → Map- Example: Conjugacy Classes of SL(2, 3)
- Subgroups
- Definition of Subgroups by Generators
- Membership and Coercion
- Inclusion and Equality
S subset G: { GrpPCElt}, GrpPC → BoolEltS notsubset G: { GrpPCElt}, GrpPC → BoolEltH subset G: GrpPC, GrpPC → BoolEltIsSubgroup(H,G): GrpPC, GrpPC → BoolEltH notsubset G: GrpPC, GrpPC → BoolEltG eq H: GrpPC, GrpPC → BoolEltG ne H: GrpPC, GrpPC → BoolEltInclusionMap(G, H): GrpPC, GrpPC → Map
- Standard Subgroup Constructions
H ^ g: GrpPC, GrpPCElt → GrpPCConjugate(H, g): GrpPC, GrpPCElt → GrpPCH meet K: GrpPC, GrpPC → GrpPCH meet:= K: GrpPC, GrpPC → GrpPCCommutatorSubgroup(G, H, K): GrpPC, GrpPC, GrpPC → GrpPCCommutatorSubgroup(H, K): GrpPC, GrpPC → GrpPCCentralizer(G, g): GrpPC, GrpPCElt → GrpPCCentraliser(G, g): GrpPC, GrpPCElt → GrpPCCentralizer(G, H): GrpPC, GrpPC → GrpPCCentraliser(G, H): GrpPC, GrpPC → GrpPCCore(G, H): GrpPC, GrpPC → GrpPCH ^ G: GrpPC, GrpPC → GrpPCNormalClosure(G, H): GrpPC, GrpPC → GrpPCNormalizer(G, H): GrpPC, GrpPC → GrpPCNormaliser(G, H): GrpPC, GrpPC → GrpPC- Example: Subgroup Constructions
- Properties of Subgroups
- Predicates for Subgroups
IsCentral(G, H): GrpPC, GrpPC → BoolEltIsConjugate(G, H, K): GrpPC, GrpPC, GrpPC → BoolElt, GrpPCEltIsMaximal(G, H): GrpPC, GrpPC → BoolEltIsNormal(G, H): GrpPC, GrpPC → BoolEltIsSelfNormalizing(G, H): GrpPC, GrpPC → BoolEltIsSubnormal(G, H): GrpPC, GrpPC → BoolElt- Example: Predicates for Subgroups
- Hall \(\pi\)-Subgroups and Sylow Systems
ComplementBasis(G): GrpPC → [GrpPC]HallSubgroup(G, S): GrpPC, { RngIntElt} → GrpPCHallSubgroup(G, S): GrpPC, RngIntElt → GrpPCpCore(G, S): GrpPC, { RngIntElt} → GrpPCpCore(G, S): GrpPC, RngIntElt → GrpPCSylowBasis(G): GrpPC → [GrpPC]SylowSubgroup(G, p): GrpPC, RngIntElt → GrpPCSylow(G, p): GrpPC, RngIntElt → GrpPCSystemNormalizer(G): GrpPC → GrpPCSystemNormaliser(G): GrpPC → GrpPC- Example: Hall Subgroups of a Wreath Product
- Conjugacy Classes of Subgroups
- Quotient Groups
- Normal Subgroups and Subgroup Series
- Cosets
- Coset Tables and Transversals
Transversal(G, H): GrpPC, GrpPC → {@ GrpPCElt @}, MapRightTransversal(G, H): GrpPC, GrpPC → {@ GrpPCElt @}, MapCosetTable(G, H): GrpPC, GrpPC → MapTransversal(G, H, K): GrpPC, GrpPC, GrpPC → {@ GrpPCElt @}, MapShortCosets(p, H, G): GrpPCElt, GrpPC, GrpPC → [GrpPCElt]
- Action on a Coset Space
- Automorphism Group
- Generating \(p\)-groups
GeneratepGroups(p, d, c : parameters): RngIntElt, RngIntElt, RngIntElt → [GrpPC], RngIntEltDescendants(G : parameters): GrpPC → [GrpPC], RngIntEltDescendants(G, c : parameters): GrpPC, RngIntElt → [GrpPC], RngIntElt- Example: Descendants of a 2-Group
- Example: 2-Generator Groups of Exponent 4
- Example: 2-Generator 3-Groups of Abundance Zero
ClassTwo(p, d : parameters): RngIntElt, RngIntElt → SeqEnumClassTwo(p, d, Step : parameters): RngIntElt, RngIntElt, SeqEnum → SeqEnumClassTwo(p, d, s : parameters): RngIntElt, RngIntElt, RngIntElt → RngIntElt- Example: Counting p-Class 2 Groups
- Representation Theory
CharacterDegrees(G): GrpPC → [ Tup ]CharacterDegrees(G, z, p): GrpPC, GrpPCElt, RngIntElt → [ Tup ]CharacterDegrees(G): GrpFin → [ Tup ]CharacterDegreesPGroup(G): GrpFin → [ RngIntElt ]CharacterTable(G: parameters): GrpPC → TabChtrCharacterTableConlon(G): GrpPC → [ AlgChtrElt ]GModule(G, M): GrpPC, AlgMat → ModAlgGModule(G, A): GrpPC, GrpPC → ModAlg, MapGModule(G, A, B): GrpPC, GrpPC, GrpPC → ModAlg, MapAbsolutelyIrreducibleRepresentationsSchur(G, k: parameters): GrpPC, Rng → List[Map]AbsolutelyIrreducibleModulesSchur(G, k: parameters): GrpPC, Rng → List[GModule]AbsolutelyIrreducibleRepresentationsSchur(G, k, i: parameters): GrpPC, Rng, RngIntElt → List[Map]AbsolutelyIrreducibleModulesSchur(G, k, i: parameters): GrpPC, Rng, RngIntElt → List[GModule]AbsolutelyIrreducibleRepresentationsSchur(G, k, L: parameters): GrpPC, Rng, List[Map] → List[Map]AbsolutelyIrreducibleModulesSchur(G, k, L: parameters): GrpPC, Rng, List[GModule] → List[GModule]AbsolutelyIrreducibleRepresentationsSchur(G, k, L, i: parameters): GrpPC, Rng, List[Map], RngIntElt → List[Map]AbsolutelyIrreducibleModulesSchur(G, k, L, i: parameters): GrpPC, Rng, RngIntElt → List[GModule]IrreducibleRepresentationsSchur(G, k: parameters): GrpPC, Rng → List[Map]IrreducibleModulesSchur(G, k: parameters): GrpPC, Rng → List[GModule]IrreducibleRepresentationsSchur(G, k, i: parameters): GrpPC, Rng, RngIntElt → List[Map]IrreducibleModulesSchur(G, k, i: parameters): GrpPC, Rng, RngIntElt → List[GModule]IrreducibleRepresentationsSchur(G, k, L: parameters): GrpPC, Rng, List[Map] → List[Map]IrreducibleModulesSchur(G, k, L: parameters): GrpPC, Rng, List[GModule] → List[GModule]IrreducibleRepresentationsSchur(G, k, L, i: parameters): GrpPC, Rng, List[Map], RngIntElt → List[Map]IrreducibleModulesSchur(G, k, L, i: parameters): GrpPC, Rng, List[GModule], RngIntElt → List[GModule]- Example: Representations of the Dihedral Group of Order 20
- Central Extensions
ExtGenerators(G, U): GrpPC, GrpPC → [<AlgMatElt, RngIntElt>]HomGenerators(G, U): GrpPC, GrpPC → [<AlgMatElt, RngIntElt>]ElementSequence(G): GrpPC → SeqEnumRepresentativeCocycles(G, U, Ext, Hom): GrpPC, GrpPC, [AlgMatElt], [AlgMatElt] → [AlgMatElt]CentralExtension(G, U, A): GrpPC, GrpPC, AlgMatElt → GrpPCCentralExtensions(G, U, Q): GrpPC, GrpPC, [AlgMatElt] → [GrpPC]CentralExtensionProcess(G, U): GrpPC, GrpPC → ProcNextExtension(~P): Rec → GrpPCIsEmpty(P): Rec → BoolElt- Example: Central Extensions of a Dihedral Group
- Transfer Between Group Categories
- More About Presentations
- Conditioned Presentations
- Special Presentations
SpecialPresentation(G): GrpPC → GrpPCSpecialWeights(G): GrpPC → [ <RngIntElt, RngIntElt, RngIntElt> ]NilpotentLength(G): GrpPC → RngIntEltNilpotentBoundary(G,i): GrpPC, RngIntElt → RngIntEltMinorLength(G,i): GrpPC, RngIntElt → RngIntEltMinorBoundary(G,i,j): GrpPC, RngIntElt, RngIntElt → RngIntEltLayerLength(G,i,j): GrpPC, RngIntElt, RngIntElt → RngIntEltLayerBoundary(G,i,j,k): GrpPC, RngIntElt, RngIntElt, RngIntElt → RngIntElt- Example: User and Special Presentations
- CompactPresentation
- Optimizing Magma Code
- \(p\)-Groups of Tame Genus
- Verbose Printing
- Constructors
TGRandomGroup(q, n, g : parameters): RngIntElt, RngIntElt, RngIntElt → GrpPC- Example: Random Groups of Tame Genus
RandomGenus2Group(q, d : parameters): RngIntElt, [RngIntElt] → GrpPC- Example: Genus 2 Groups with Prescribed Blocks
RandomGenus1Group(q, d, r : parameters): RngIntElt, RngIntElt, RngIntElt → GrpPC- Example: Heisenberg Groups
Genus2Group(f): RngUPolElt → GrpPCGenus2Group(f): RngMPolElt → GrpPC- Example: Genus 2 Group from a Pfaffian
- Direct Indecomposability
- Genus
- Isomorphism
- Automorphism Groups
- Canonical Labelling
- Bibliography