Cohomology and Extensions
- Introduction
- Creation of a Cohomology Module
CohomologyModule(G, M): GrpPerm, ModGrp → ModCohoCohomologyModule(G, M): GrpPC, ModGrp → ModCohoCohomologyModule(G, M): GrpMat, ModGrp → ModCohoCohomologyModule(G, M): GrpFP, ModGrp → ModCohoCohomologyModule(G, Q, T): GrpPerm, SeqEnum, SeqEnum → ModCohoCohomologyModule(G, Q, T): GrpPC, SeqEnum, SeqEnum → ModCohoCohomologyModule(G, Q, T): GrpMat, SeqEnum, SeqEnum → ModCohoCohomologyModule(G, Q, T): GrpFP, SeqEnum, SeqEnum → ModCoho- Example: Cohomology Module for PSL(3, 2)
- Example: Cohomology Module for ASL(3, 5)
- Example: Cohomology Module for a Cyclic Group
- Example: Cohomology Module for an Infinite Group
CohomologyModule(G, A, M): GrpPerm, GrpAb, Any → ModCoho
- Accessing Properties of the Cohomology Module
- Calculating Cohomology
CohomologyGroup(CM, n): ModCoho, RngIntElt → ModTupRng- Example: Cohomology for a Cyclic Group of Order 4
- Example: Cohomology of an Infinite FP-Group
CohomologicalDimension(CM, n): ModCoho, RngIntElt → RngIntEltCohomologicalDimension(M, n): ModGrp, RngIntElt → RngIntEltCohomologicalDimensions(M, n): ModGrp, RngIntElt → RngIntEltCohomologicalDimension(G, M, n): GrpPerm, ModRng, RngIntElt → RngIntEltH1Dimension(F, f, M): GrpFP, Map, ModGrp → RngIntEltH1Dimension(G, f, K): GrpFP, Map, Rng → RngIntEltH1DimensionSymmetricSquare(G, f, K): GrpFP, Map, Rng → RngIntEltH1DimensionExteriorSquare(G, f, K): GrpFP, Map, Rng → RngIntElt- Example: Cohomology of A8
- Example: Cohomology of ASL(3, 5)
- Example: Cohomology Where the Old Method Fails
- Cocycles
ZeroCocycle(CM, s): ModCoho, SeqEnum → UserProgramZeroCocycle(CM, s): ModCoho, ModTupRngElt → UserProgramIdentifyZeroCocycle(CM, s): ModCoho, UserProgram → ModTupRngEltOneCocycle(CM, s): ModCoho, SeqEnum → UserProgramOneCocycle(CM, s): ModCoho, ModTupRngElt → UserProgramIdentifyOneCocycle(CM, s): ModCoho, UserProgram → ModTupRngEltIsOneCoboundary(CM, s): ModCoho, UserProgram → BoolElt, UserProgramTwoCocycle(CM, s): ModCoho, SeqEnum → UserProgramTwoCocycle(CM, s): ModCoho, ModTupRngElt → UserProgramIdentifyTwoCocycle(CM, s): ModCoho, UserProgram → ModTupRngEltIsTwoCoboundary(CM, s): ModCoho, UserProgram → BoolElt, UserProgram- Example: Cocycles for a Cyclic Group
- The Restriction to a Subgroup
- Other Operations on Cohomology Modules
CorestrictionMapImage(G, C, c, i): Grp, ModCoho, UserProgram, RngIntElt → UserProgramCorestrictCocycle(G, C, c, i): Grp, ModCoho, UserProgram, RngIntElt → UserProgramInflationMapImage(M, c): Map, UserProgram → UserProgramLiftCocycle(M, c): Map, UserProgram → UserProgramCoboundaryMapImage(M, i, c): ModCoho, RngIntElt, UserProgram → UserProgram
- Constructing Extensions
Extension(CM, s): ModCoho, SeqEnum → Grp, HomGrp, MapExtension(CM, s): ModCoho, ModTupRngElt → Grp, HomGrp, MapExtension(GrpPerm, CM, s): Cat, ModCoho, SeqEnum → GrpPerm, HomGrp, MapExtension(GrpPerm, CM, s): Cat, ModCoho, ModTupRngElt → GrpPerm, HomGrp, Map- Example: Schur Cover of A7
- Example: Dempwolff Group Using a Chosen Subgroup
SplitExtension(CM): ModCoho → Grp, HomGrp, MapSplitExtension(G, M): ModCoho → Grp, ModGrp -> HomGrp, MapSplitExtension(M): ModCoho → ModGrp -> HomGrp, MapSplitExtension(GrpPerm,CM): Cat, ModCoho → GrpPerm, HomGrp, MapSplitExtension(GrpPerm,G,M): Cat, ModCoho → GrpPerm, HomGrp, MapSplitExtension(GrpPerm,M): Cat, ModCoho → GrpPerm, HomGrp, Map- Example: Split Extension Using Module Elements
- Example: Split Extension Using Cosets
pMultiplicator(G, p): GrpPerm, RngIntElt → [ RngIntElt ]pCover(G, F, p): GrpPerm, GrpFP, RngIntElt → GrpFP- Example: Non-Split Extension by A8
- Example: Dempwolff Group from a Core-Free Subgroup
- Example: Cohomology of an Integral Permutation Module
- Constructing Distinct Extensions
- Finite Group Cohomology
- Creation of Gamma-groups
GammaGroup(Gamma, A, action): Grp, Grp, Map[Grp, GrpAuto] → GGrpInducedGammaGroup(A, B): GGrp, Grp → GGrp- Example: Creating a Gamma-Group
IsNormalised(B, action): Grp, Map → BoolEltIsInduced(AmodB): GGrp → BoolElt, GGrp, GGrp, Map, Map
- Accessing Information
- One Cocycles
OneCocycle(A, imgs): GGrp, SeqEnum[GrpElt] → OneCoCOneCocycle(A, alpha): GGrp, Map[Grp,Grp] → OneCoCTrivialOneCocycle(A): GGrp → OneCoCIsOneCocycle(A, imgs): GGrp, SeqEnum[GrpElt] → BoolElt, OneCoCIsOneCocycle(A, alpha): GGrp, Map[Grp,Grp] → BoolElt, OneCoCAreCohomologous(alpha, beta): OneCoC, OneCoC → BoolElt, GrpEltCohomologyClass(alpha): OneCoC → SetIndx[OneCoC]InducedOneCocycle(AmodB, alpha): GGrp, OneCoC → OneCoCInducedOneCocycle(A, B, alpha): GGrp, Grp, OneCoC → OneCoCExtendedOneCocycle(alpha): OneCoC → SetEnum[OneCoC]ExtendedCohomologyClass(alpha): OneCoC → SetEnum[OneCoC]GammaGroup(alpha): OneCoC → GGrpCocycleMap(alpha): OneCoC → Map
- Group Cohomology
- Bibliography