- Introduction
- Construction of Elements
- Construction of an Element
- Coercion
- Homomorphisms
- Arithmetic with Elements
- g * h : GrpElt, GrpElt -> GrpElt
- g ^ n : GrpElt, RngIntElt -> GrpElt
- g / h : GrpElt, GrpElt -> GrpElt
- g ^ h : GrpElt, GrpElt -> GrpElt
- (g, h) : GrpElt, GrpElt -> GrpElt
- (g1, ..., gr) : GrpElt, ..., GrpElt -> GrpElt
- g eq h : GrpElt, GrpElt -> BoolElt
- g ne h : GrpElt, GrpElt -> BoolElt
- IsId(g) : GrpElt -> BoolElt
- Order(g) : GrpElt -> RngIntElt
- Example Grp_Arithmetic (H63E3)
- Construction of a General Group
- Standard Groups and Extensions
- Construction of a Standard Group
- AbelianGroup(C, Q) : Cat, [ RngIntElt ] -> GrpFin
- AlternatingGroup(C, n) : Cat, RngIntElt -> GrpFin
- CyclicGroup(C, n) : Cat, RngIntElt -> GrpFin
- DihedralGroup(C, n) : Cat, RngIntElt -> GrpFin
- DicyclicGroup(n) : RngIntElt -> GrpFP
- SymmetricGroup(C, n) : Cat, RngIntElt -> GrpFin
- ExtraSpecialGroup(C, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpFin
- Example Grp_StandardGroups (H63E8)
- Construction of Extensions
- DirectProduct(G, H) : Grp, Grp -> Grp
- DirectProduct(Q) : [ Grp ] -> Grp
- SemidirectProduct(K, H, f: parameters) : Grp, Grp, Map -> Grp, Map, Map, Map
- Example Grp_semidirect (H63E9)
- AffineSplitExtension(M: parameters) : ModGrp -> Grp, Map, Map, Map
- Example Grp_affine-split (H63E10)
- Example Grp_Extensions (H63E11)
- Transfer Functions Between Group Categories
- pQuotient(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map
- CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, Grp
- CosetImage(G, H) : Grp, Grp -> GrpPerm
- CosetKernel(G, H) : Grp, Grp -> Grp
- MinimalDegreePermutationRepresentation(G: parameters) : Grp -> Hom(Grp), GrpPerm
- Example Grp_MinimalDegreePermutationRepresentation (H63E12)
- GPCGroup(G) : Grp -> GrpGPC, Hom(Grp)
- PCGroup(G) : Grp -> GrpPC, Hom(Grp)
- FPGroup(G: parameters) : GrpPerm :-> GrpFP, Hom(Grp)
- Example Grp_CosetAction (H63E13)
- Example Grp_CosetAction-2 (H63E14)
- Example Grp_FPGroup (H63E15)
- Basic Operations
- Operations on the Set of Elements
- Order and Index Functions
- Membership and Equality
- Set Operations
- Random Elements
- Action on a Coset Space
- CosetTable(G, H) : GrpFin, GrpFin -> Map
- [Future release] CosetTable(G, f) : GrpFin, Hom(GrpFin) -> Hom(GrpFin)
- Transversal(G, H) : Grp, Grp -> {@ GrpElt @}, Map
- CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, Grp
- CosetImage(G, H) : Grp, Grp -> GrpPerm
- CosetKernel(G, H) : Grp, Grp -> Grp
- Standard Subgroup Constructions
- H ^ g : GrpFin, GrpFinElt -> GrpFin
- H meet K : GrpFin, GrpFin -> GrpFin
- CommutatorSubgroup(G, H, K) : GrpFin, GrpFin, GrpFin -> GrpFin
- Centralizer(G, g) : GrpFin, GrpFinElt -> GrpFin
- Centralizer(G, H) : GrpFin, GrpFin -> GrpFin
- Core(G, H) : GrpFin, GrpFin -> GrpFin
- H ^ G : GrpFin, GrpFin -> GrpFin
- Normalizer(G, H) : GrpFin, GrpFin -> GrpFin
- pCore(G, p) : GrpFin, RngIntElt -> GrpFin
- SylowSubgroup(G, p) : GrpFin, RngIntElt -> GrpFin
- Abstract Group Predicates
- IsAbelian(G) : GrpFin -> BoolElt
- IsCyclic(G) : GrpFin -> BoolElt
- IsElementaryAbelian(G) : GrpFin -> BoolElt
- IsCentral(G, H) : GrpFin, GrpFin -> BoolElt
- IsConjugate(G, g, h) : GrpFin, GrpFinElt, GrpFinElt -> BoolElt, GrpFinElt
- IsConjugate(G, H, K) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt
- IsExtraSpecial(G) : GrpFin -> BoolElt
- IsHyperelementary(G) : Grp -> BoolElt,Grp,Grp
- Example Grp_grp-ishyperelementary (H63E23)
- IsMaximal(G, H) : GrpFin, GrpFin -> BoolElt
- IsNilpotent(G) : GrpFin -> BoolElt
- IsNormal(G, H) : GrpFin, GrpFin -> BoolElt
- IsPerfect(G) : GrpFin -> BoolElt
- IsQGroup(G) : Grp -> BoolElt
- Example Grp_grp-isqgroup (H63E24)
- IsSelfNormalizing(G, H) : GrpFin, GrpFin -> BoolElt
- IsSimple(G) : GrpFin -> BoolElt
- IsSoluble(G) : GrpFin -> BoolElt
- IsSpecial(G) : GrpFin -> BoolElt
- IsSubnormal(G, H) : GrpFin, GrpFin -> BoolElt
- IsTrivial(G) : Grp -> BoolElt
- Characteristic Subgroups and Normal Structure
- Conjugacy Classes of Elements
- Class(H, x) : GrpFin, GrpFinElt -> { GrpFinElt }
- ClassMap(G: parameters) : GrpFin -> Map
- ConjugacyClasses(G: parameters) : GrpFin -> [ <RngIntElt, RngIntElt, GrpFinElt> ]
- ClassesData(G: parameters) : GrpFin -> [ <RngIntElt, RngIntElt> ]
- ClassRepresentative(G, x) : GrpFin, GrpFinElt -> GrpFinElt
- IsConjugate(G, g, h) : GrpFin, GrpFinElt, GrpFinElt -> BoolElt, GrpFinElt
- IsConjugate(G, H, K) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt
- Exponent(G) : GrpFin -> RngIntElt
- NumberOfClasses(G) : GrpFin -> RngIntElt
- PowerMap(G) : GrpFin -> Map
- Example Grp_Classes (H63E25)
- Conjugacy Classes of Subgroups
- Conjugacy Classes of Subgroups
- SubgroupClasses(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- ElementaryAbelianSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- AbelianSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- CyclicSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- NilpotentSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- SolubleSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- NonsolvableSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- PerfectSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- SimpleSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- RegularSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
- SetVerbose("SubgroupLattice", i) : MonStgElt, RngIntElt ->
- Class(G, H) : GrpFin, GrpFin -> { GrpFin }
- Example Grp_Subgroups (H63E26)
- The Poset of Subgroup Classes
- All Subgroups and Intermediate Subgroups
- Cohomology
- pMultiplicator(G, p) : GrpFin, RngIntElt -> [ RngIntElt ]
- pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFinFP
- CohomologicalDimension(G, M, i) : GrpFin, ModRng, RngIntElt -> RngIntElt
- ExtensionProcess(G, M, F) : GrpPerm, ModRng, GrpFP -> GrpFPExtProc
- Extension(P, Q) : Process -> GrpFinFP
- SplitExtension(G, M, F) : GrpPerm, ModRng, GrpFP -> GrpFP
- Characters and Representations
- Character Theory
- Representation Theory
- GModule(G, S) : GrpFin, AlgMat -> ModGrpFin
- GModule(G, A, B) : GrpFin, GrpFin, GrpFin -> ModGrpFin, Map
- PermutationModule(G, H, R) : GrpFin, GrpFin, Rng -> ModGrpFin
- PermutationModule(G, R) : GrpPerm, Rng -> ModGrpFin
- Example Grp_Modules (H63E30)
- Example Grp_Modules-2 (H63E31)
- Databases of Groups
- Bibliography
V2.28, 13 July 2023