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Acknowledgements
Introduction
Creation of a Cohomology Module
Accessing Properties of the Cohomology Module
Calculating Cohomology
Cocycles
The restriction to a subgroup
Other operations on cohomology modules
Constructing Extensions
Constructing Distinct Extensions
Finite Group Cohomology
Creation of Gamma-groups
Accessing Information
One Cocycles
Group Cohomology
Bibliography
DETAILS
Introduction
Creation of a Cohomology Module
CohomologyModule(G, M) : GrpPerm, ModGrp -> ModCoho
CohomologyModule(G, Q, T) : GrpPerm, SeqEnum, SeqEnum -> ModCoho
Example GrpCoh_coho-module1 (H59E1)
CohomologyModule(G, A, M) : GrpPerm, GrpAb, Any -> ModCoho
Accessing Properties of the Cohomology Module
Module(CM) : ModCoho -> ModGrp
Invariants(CM) : ModCoho -> SeqEnum
Dimension(CM) : ModCoho -> RngIntElt
Ring(CM) : ModCoho -> ModGrp
Group(CM) : ModCoho -> Grp
FPGroup(CM) : ModCoho -> Grp, HomGrp
MatrixOfElement(CM, g) : ModCoho, GrpElt -> AlgMatElt
Calculating Cohomology
CohomologyGroup(CM, n) : ModCoho, RngIntElt -> ModTupRng
CohomologicalDimension(CM, n) : ModCoho, RngIntElt -> RngIntElt
CohomologicalDimension(G, M, n) : GrpPerm, ModRng, RngIntElt -> RngIntElt
Example GrpCoh_coho-example (H59E2)
Example GrpCoh_more-difficult (H59E3)
Cocycles
ZeroCocycle(CM, s) : ModCoho, SeqEnum -> UserProgram
IdentifyZeroCocycle(CM, s) : ModCoho, UserProgram -> ModTupRngElt
OneCocycle(CM, s) : ModCoho, SeqEnum -> UserProgram
IdentifyOneCocycle(CM, s) : ModCoho, UserProgram -> ModTupRngElt
IsOneCoboundary(CM, s) : ModCoho, UserProgram -> BoolElt, UserProgram
TwoCocycle(CM, s) : ModCoho, SeqEnum -> UserProgram
IdentifyTwoCocycle(CM, s) : ModCoho, UserProgram -> ModTupRngElt
IsTwoCoboundary(CM, s) : ModCoho, UserProgram -> BoolElt, UserProgram
Example GrpCoh_cocycles (H59E4)
The restriction to a subgroup
Restriction(CM, H) : ModCoho, Grp -> ModCoho
Example GrpCoh_restriction (H59E5)
Other operations on cohomology modules
CorestrictionMapImage(G, C, c, i):Grp, ModCoho, UserProgram, RngIntElt -> UserProgram
InflationMapImage(M, c) : Map, UserProgram -> UserProgram
CoboundaryMapImage(M, i, c) : ModCoho, RngIntElt, UserProgram -> UserProgram
Constructing Extensions
Extension(CM, s) : ModCoho, SeqEnum -> Grp, HomGrp, Map
SplitExtension(CM) : ModCoho -> Grp, HomGrp, Map
pMultiplicator(G, p) : GrpPerm, RngIntElt -> [ RngIntElt ]
pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFP
Example GrpCoh_straightforward (H59E6)
Example GrpCoh_module-integers (H59E7)
Constructing Distinct Extensions
DistinctExtensions(CM) : ModCoho -> SeqEnum
Example GrpCoh_distinct-extensions (H59E8)
ExtensionsOfElementaryAbelianGroup(p, d, G) : RngIntElt, RngIntElt, GrpPerm -> SeqEnum
Example GrpCoh_extensions-abelian (H59E9)
ExtensionsOfSolubleGroup(H, G) : GrpPerm, GrpPerm -> SeqEnum
Example GrpCoh_extensions-soluble (H59E10)
Example GrpCoh_distinct-extensions (H59E11)
IsExtensionOf(G) : GrpPerm -> [],
IsExtensionOf(L) : [GrpPerm] -> [], []
Finite Group Cohomology
Creation of Gamma-groups
GammaGroup(Gamma, A, action) : Grp, Grp, Map[Grp, GrpAuto] -> GGrp
InducedGammaGroup(A, B) : GGrp, Grp -> GGrp
Example GrpCoh_createGGrp (H59E12)
IsNormalised(B, action) : Grp, Map -> BoolElt
IsInduced(AmodB) : GGrp -> BoolElt, GGrp, GGrp, Map, Map
Accessing Information
Group(A) : GGrp -> Grp
GammaAction(A) : GGrp -> Map[Grp, GrpAuto]
ActingGroup(A) : GGrp -> Grp
One Cocycles
OneCocycle(A, imgs) : GGrp, SeqEnum[GrpElt] -> OneCoC
TrivialOneCocycle(A) : GGrp -> OneCoC
IsOneCocycle(A, imgs) : GGrp, SeqEnum[GrpElt] -> BoolElt, OneCoC
AreCohomologous(alpha, beta) : OneCoC, OneCoC -> BoolElt, GrpElt
CohomologyClass(alpha) : OneCoC -> SetIndx[OneCoC]
InducedOneCocycle(AmodB, alpha) : GGrp, OneCoC -> OneCoC
ExtendedOneCocycle(alpha) : OneCoC -> SetEnum[OneCoC]
ExtendedCohomologyClass(alpha) : OneCoC -> SetEnum[OneCoC]
GammaGroup(alpha) : OneCoC -> GGrp
CocycleMap(alpha) : OneCoC -> Map
Group Cohomology
Cohomology(A, n) : GGrp, RngIntElt -> SetEnum[OneCoC]
OneCohomology(A) : GGrp -> SetEnum[OneCoC]
TwistedGroup(A, alpha) : GGrp, OneCoC -> GGrp
Bibliography
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