- Introduction
- Polycyclic Groups and Polycyclic Presentations
- Introduction
- Specification of Elements
- Access Functions for Elements
- Arithmetic Operations on Elements
- g * h : GrpGPCElt, GrpGPCElt -> GrpGPCElt
- g *:= h : GrpGPCElt, GrpGPCElt ->
- g ^ n: GrpGPCElt, RngIntElt -> GrpGPCElt
- g ^:= n: GrpGPCElt, RngIntElt ->
- g / h : GrpGPCElt, GrpGPCElt -> GrpGPCElt
- g /:= h : GrpGPCElt, GrpGPCElt ->
- g ^ h : GrpGPCElt, GrpGPCElt -> GrpGPCElt
- g ^:= 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] -> GrpGPC
- CyclicGroup(GrpGPC, n) : Cat, RngIntElt -> GrpGPC
- DihedralGroup(GrpGPC, n) : Cat, RngIntElt -> GrpGPC
- ElementaryAbelianGroup(GrpGPC, p, n) : Cat, RngIntElt, RngIntElt -> GrpGPC
- ExtraSpecialGroup(GrpGPC, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpGPC
- FreeAbelianGroup(GrpGPC, n) : Cat, RngIntElt -> GrpGPC
- FreeNilpotentGroup(r, e) : RngIntElt, RngIntElt -> GrpGPC
- Example GrpGPC_Homomorphism (H79E4)
- Example GrpGPC_Symmetric2 (H79E5)
- Conversion between Categories
- Access Functions for Groups
- Set-Theoretic Operations in a Group
- Coset Spaces
- CosetTable(G, H) : GrpGPC, GrpGPC -> Map
- Transversal(G, H) : GrpGPC, GrpGPC -> {@ GrpGPCElt @}, Map
- Example GrpGPC_CosetTable (H79E7)
- CosetAction(G, H) : GrpGPC, GrpGPC -> Map, GrpPerm, GrpGPC
- CosetImage(G, H) : GrpGPC, GrpGPC -> GrpPerm
- CosetKernel(G, H) : GrpGPC, GrpGPC -> GrpGPC
- Example GrpGPC_CosetAction (H79E8)
- The Subgroup Structure
- General Subgroup Constructions
- Subgroup Constructions Requiring a Nil-po-tent Covering Group
- H meet K : GrpGPC, GrpGPC -> GrpGPC
- H meet:= K : GrpGPC, GrpGPC -> GrpGPC
- Centraliser(G, g) : GrpGPC, GrpGPCElt -> GrpGPC
- Centraliser(G, H) : GrpGPC, GrpGPC -> GrpGPC
- Core(G, H) : GrpGPC, GrpGPC -> GrpGPC
- Normaliser(G, H) : GrpGPC, GrpGPC -> GrpGPC
- General Group Properties
- Normal Structure and Characteristic Subgroups
- Conjugacy
- IsConjugate(G, g, h) : GrpGPC, GrpGPCElt, GrpGPCElt -> BoolElt, GrpGPCElt
- IsConjugate(G, H, K) : GrpGPC, GrpGPC, GrpGPC -> BoolElt, GrpGPCElt
- Example GrpGPC_Conjugacy (H79E12)
- Representation Theory
- EFAModuleMaps(G) : GrpGPC -> [ModGrp]
- EFAModules(G) : GrpGPC -> [ModGrp]
- GModule(G, A, p) : GrpGPC, GrpGPC, RngIntElt -> ModGrp, Map
- GModule(G, A, B, p) : GrpGPC, GrpGPC, GrpGPC, RngIntElt -> ModGrp, Map
- GModulePrimes(G, A) : GrpGPC, GrpGPC -> SetMulti
- GModulePrimes(G, A, B) : GrpGPC, GrpGPC, GrpGPC -> SetMulti
- SemisimpleEFAModuleMaps(G) : GrpGPC -> [ModGrp]
- SemisimpleEFAModules(G) : GrpGPC -> [ModGrp]
- Example GrpGPC_RepresentationTheory (H79E13)
- Example GrpGPC_gmoduleprimes (H79E14)
- Example GrpGPC_FittingSubgroup (H79E15)
- Example GrpGPC_ModuleMaps (H79E16)
- Power Groups
- Bibliography
V2.28, 13 July 2023