[Next][Prev] [_____] [Left] [Up] [Index] [Root]

COHOMOLOGY AND EXTENSIONS

 
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 (H68E1)
      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(M, n) : ModGrp, n -> RngIntElt
      CohomologicalDimensions(M, n) : ModGrp, n -> RngIntElt
      CohomologicalDimension(G, M, n) : GrpPerm, ModRng, RngIntElt -> RngIntElt
      Example GrpCoh_coho-example (H68E2)
      Example GrpCoh_more-difficult (H68E3)

 
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 (H68E4)

 
The Restriction to a Subgroup
      Restriction(CM, H) : ModCoho, Grp -> ModCoho
      Example GrpCoh_restriction (H68E5)

 
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 (H68E6)
      Example GrpCoh_module-integers (H68E7)

 
Constructing Distinct Extensions
      DistinctExtensions(CM) : ModCoho -> SeqEnum
      Example GrpCoh_distinct-extensions (H68E8)
      ExtensionsOfElementaryAbelianGroup(p, d, G) : RngIntElt, RngIntElt, GrpPerm -> SeqEnum
      Example GrpCoh_extensions-abelian (H68E9)
      ExtensionsOfSolubleGroup(H, G) : GrpPerm, GrpPerm -> SeqEnum
      Example GrpCoh_extensions-soluble (H68E10)
      Example GrpCoh_distinct-extensions (H68E11)
      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 (H68E12)
            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
            Example GrpCoh_large example (H68E13)

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Mon Dec 17 14:40:36 EST 2012