Polycyclic Groups
- Introduction
- Polycyclic Groups and Polycyclic Presentations
- Introduction
- Specification of Elements
- Access Functions for Elements
- Arithmetic Operations on Elements
g * h: GrpGPCElt, GrpGPCElt → GrpGPCEltg *:= h: GrpGPCElt, GrpGPCEltg ^ n: GrpGPCElt, RngIntElt → GrpGPCEltg ^:= n: GrpGPCElt, RngIntEltg / h: GrpGPCElt, GrpGPCElt → GrpGPCEltg /:= h: GrpGPCElt, GrpGPCEltg ^ h: GrpGPCElt, GrpGPCElt → GrpGPCEltg ^:= h: GrpGPCElt, GrpGPCElt(g₁, ..., gₙ): List(GrpGPCElt) → GrpGPCElt
- Operators for Elements
- Comparison Operators for Elements
- Specification of a Polycyclic Presentation
- Properties of a Polycyclic Presentation
- Subgroups, Quotient Groups, Homomorphisms and Extensions
- Construction of Subgroups
- Coercions Between Groups and Subgroups
- Construction of Quotient Groups
- Homomorphisms
- Construction of Extensions
- Construction of Standard Groups
AbelianGroup(GrpGPC, Q): Cat, [RngIntElt] → GrpGPCCyclicGroup(GrpGPC, n): Cat, RngIntElt → GrpGPCDihedralGroup(GrpGPC, n): Cat, RngIntElt → GrpGPCElementaryAbelianGroup(GrpGPC, p, n): Cat, RngIntElt, RngIntElt → GrpGPCExtraSpecialGroup(GrpGPC, p, n : parameters): Cat, RngIntElt, RngIntElt → GrpGPCFreeAbelianGroup(GrpGPC, n): Cat, RngIntElt → GrpGPCFreeNilpotentGroup(r, e): RngIntElt, RngIntElt → GrpGPCExample: HomomorphismExample: Symmetric2
- Conversion between Categories
- Access Functions for Groups
- Set-Theoretic Operations in a Group
- Functions Relating to Group Order
- Membership and Equality
g in G: GrpGPCElt, GrpGPC → BoolEltg notin G: GrpGPCElt, GrpGPC → BoolEltS subset G: { GrpGPCElt}, GrpGPC → BoolEltS notsubset G: { GrpGPCElt}, GrpGPC → BoolEltH subset G: GrpGPC, GrpGPC → BoolEltIsSubgroup(H,G): GrpGPC, GrpGPC → BoolEltH notsubset G: GrpGPC, GrpGPC → BoolEltG eq H: GrpGPC, GrpGPC → BoolEltG ne H: GrpGPC, GrpGPC → BoolElt
- Set Operations
- Coset Spaces
CosetTable(G, H): GrpGPC, GrpGPC → MapTransversal(G, H): GrpGPC, GrpGPC → { @ GrpGPCElt @}, MapRightTransversal(G, H): GrpGPC, GrpGPC → { @ GrpGPCElt @}, MapExample: Coset TableCosetAction(G, H): GrpGPC, GrpGPC → Map, GrpPerm, GrpGPCCosetImage(G, H): GrpGPC, GrpGPC → GrpPermCosetKernel(G, H): GrpGPC, GrpGPC → GrpGPCExample: Coset Action
- The Subgroup Structure
- General Subgroup Constructions
H ^ g: GrpGPC, GrpGPCElt → GrpGPCConjugate(H, g): GrpGPC, GrpGPCElt → GrpGPCH ^ G: GrpGPC, GrpGPC → GrpGPCncl< G | H >: GrpGPC, GrpGPC → GrpGPCNormalClosure(G, H): GrpGPC, GrpGPC → GrpGPCCommutatorSubgroup(G, H, K): GrpGPC, GrpGPC, GrpGPC → GrpGPCCommutatorSubgroup(H, K): GrpGPC, GrpGPC → GrpGPC
- Subgroup Constructions Requiring a Nilpotent Covering Group
H meet K: GrpGPC, GrpGPC → GrpGPCH meet:= K: GrpGPC, GrpGPC → GrpGPCCentraliser(G, g): GrpGPC, GrpGPCElt → GrpGPCCentralizer(G, g): GrpGPC, GrpGPCElt → GrpGPCCentraliser(G, H): GrpGPC, GrpGPC → GrpGPCCentralizer(G, H): GrpGPC, GrpGPC → GrpGPCCore(G, H): GrpGPC, GrpGPC → GrpGPCNormaliser(G, H): GrpGPC, GrpGPC → GrpGPCNormalizer(G, H): GrpGPC, GrpGPC → GrpGPC
- General Group Properties
- Normal Structure and Characteristic Subgroups
- Conjugacy
IsConjugate(G, g, h): GrpGPC, GrpGPCElt, GrpGPCElt → BoolElt, GrpGPCEltIsConjugate(G, H, K): GrpGPC, GrpGPC, GrpGPC → BoolElt, GrpGPCEltExample: Conjugacy
- Representation Theory
EFAModuleMaps(G): GrpGPC → [ModGrp]EFAModules(G): GrpGPC → [ModGrp]GModule(G, A, p): GrpGPC, GrpGPC, RngIntElt → ModGrp, MapGModule(G, A): GrpGPC, GrpGPC → ModGrp, MapGModule(G, A, B, p): GrpGPC, GrpGPC, GrpGPC, RngIntElt → ModGrp, MapGModule(G, A, B): GrpGPC, GrpGPC, GrpGPC → ModGrp, MapGModulePrimes(G, A): GrpGPC, GrpGPC → SetMultiGModulePrimes(G, A, B): GrpGPC, GrpGPC, GrpGPC → SetMultiSemisimpleEFAModuleMaps(G): GrpGPC → [ModGrp]SemisimpleEFAModules(G): GrpGPC → [ModGrp]Example: Representation TheoryExample: gmoduleprimesExample: Fitting SubgroupExample: Module Maps
- Power Groups
