[1X8 [33X[0;0YCohomology for pcp-groups[133X[101X [33X[0;0YThe [5XGAP[105X 4 package [5XPolycyclic[105X provides methods to compute the first and second cohomology group for a pcp-group [22XU[122X and a finite dimensional [22Xℤ U[122X or [22XFU[122X module [22XA[122X where [22XF[122X is a finite field. The algorithm for determining the first cohomology group is outlined in [Eic00].[133X [33X[0;0YAs a preparation for the cohomology computation, we introduce the cohomology records. These records provide the technical setup for our cohomology computations.[133X [1X8.1 [33X[0;0YCohomology records[133X[101X [33X[0;0YCohomology records provide the necessary technical setup for the cohomology computations for polycyclic groups.[133X [1X8.1-1 CRRecordByMats[101X [33X[1;0Y[29X[2XCRRecordByMats[102X( [3XU[103X, [3Xmats[103X ) [32X function[133X [33X[0;0Ycreates an external module. Let [3XU[103X be a pcp group which acts via the list of matrices [3Xmats[103X on a vector space of the form [22Xℤ^n[122X or [22XF_p^n[122X. Then this function creates a record which can be used as input for the cohomology computations.[133X [1X8.1-2 CRRecordBySubgroup[101X [33X[1;0Y[29X[2XCRRecordBySubgroup[102X( [3XU[103X, [3XA[103X ) [32X function[133X [33X[1;0Y[29X[2XCRRecordByPcp[102X( [3XU[103X, [3Xpcp[103X ) [32X function[133X [33X[0;0Ycreates an internal module. Let [3XU[103X be a pcp group and let [3XA[103X be a normal elementary or free abelian normal subgroup of [3XU[103X or let [3Xpcp[103X be a pcp of a normal elementary of free abelian subfactor of [3XU[103X. Then this function creates a record which can be used as input for the cohomology computations.[133X [33X[0;0YThe returned cohomology record [3XC[103X contains the following entries:[133X [8X[3Xfactor[103X[8X[108X [33X[0;6Ya pcp of the acting group. If the module is external, then this is [3XPcp(U)[103X. If the module is internal, then this is [3XPcp(U, A)[103X or [3XPcp(U, GroupOfPcp(pcp))[103X.[133X [8X[3Xmats[103X[8X, [3Xinvs[103X[8X and [3Xone[103X[8X[108X [33X[0;6Ythe matrix action of [3Xfactor[103X with acting matrices, their inverses and the identity matrix.[133X [8X[3Xdim[103X[8X and [3Xchar[103X[8X[108X [33X[0;6Ythe dimension and characteristic of the matrices.[133X [8X[3Xrelators[103X[8X and [3Xenumrels[103X[8X[108X [33X[0;6Ythe right hand sides of the polycyclic relators of [3Xfactor[103X as generator exponents lists and a description for the corresponding left hand sides.[133X [8X[3Xcentral[103X[8X[108X [33X[0;6Yis true, if the matrices [3Xmats[103X are all trivial. This is used locally for efficiency reasons.[133X [33X[0;0YAnd additionally, if [22XC[122X defines an internal module, then it contains:[133X [8X[3Xgroup[103X[8X[108X [33X[0;6Ythe original group [3XU[103X.[133X [8X[3Xnormal[103X[8X[108X [33X[0;6Ythis is either [3XPcp(A)[103X or the input [3Xpcp[103X.[133X [8X[3Xextension[103X[8X[108X [33X[0;6Yinformation on the extension of [3XA[103X by [3XU/A[103X.[133X [1X8.2 [33X[0;0YCohomology groups[133X[101X [33X[0;0YLet [22XU[122X be a pcp-group and [22XA[122X a free or elementary abelian pcp-group and a [22XU[122X-module. By [22XZ^i(U, A)[122X be denote the group of [22Xi[122X-th cocycles and by [22XB^i(U, A)[122X the [22Xi[122X-th coboundaries. The factor [22XZ^i(U,A) / B^i(U,A)[122X is the [22Xi[122X-th cohomology group. Since [22XA[122X is elementary or free abelian, the groups [22XZ^i(U, A)[122X and [22XB^i(U, A)[122X are elementary or free abelian groups as well.[133X [33X[0;0YThe [5XPolycyclic[105X package provides methods to compute first and second cohomology group for a polycyclic group [3XU[103X. We write all involved groups additively and we use an explicit description by bases for them. Let [22XC[122X be the cohomology record corresponding to [22XU[122X and [22XA[122X.[133X [33X[0;0YLet [22Xf_1, ..., f_n[122X be the elements in the entry [22Xfactor[122X of the cohomology record [22XC[122X. Then we use the following embedding of the first cocycle group to describe 1-cocycles and 1-coboundaries: [22XZ^1(U, A) -> A^n : δ ↦ (δ(f_1), ..., δ(f_n))[122X[133X [33X[0;0YFor the second cohomology group we recall that each element of [22XZ^2(U, A)[122X defines an extension [22XH[122X of [22XA[122X by [22XU[122X. Thus there is a pc-presentation of [22XH[122X extending the pc-presentation of [22XU[122X given by the record [22XC[122X. The extended presentation is defined by tails in [22XA[122X; that is, each relator in the record entry [22Xrelators[122X is extended by an element of [22XA[122X. The concatenation of these tails yields a vector in [22XA^l[122X where [22Xl[122X is the length of the record entry [22Xrelators[122X of [22XC[122X. We use these tail vectors to describe [22XZ^2(U, A)[122X and [22XB^2(U, A)[122X. Note that this description is dependent on the chosen presentation in [22XC[122X. However, the factor [22XZ^2(U, A)/ B^2(U, A)[122X is independent of the chosen presentation.[133X [33X[0;0YThe following functions are available to compute explicitly the first and second cohomology group as described above.[133X [1X8.2-1 OneCoboundariesCR[101X [33X[1;0Y[29X[2XOneCoboundariesCR[102X( [3XC[103X ) [32X function[133X [33X[1;0Y[29X[2XOneCocyclesCR[102X( [3XC[103X ) [32X function[133X [33X[1;0Y[29X[2XTwoCoboundariesCR[102X( [3XC[103X ) [32X function[133X [33X[1;0Y[29X[2XTwoCocyclesCR[102X( [3XC[103X ) [32X function[133X [33X[1;0Y[29X[2XOneCohomologyCR[102X( [3XC[103X ) [32X function[133X [33X[1;0Y[29X[2XTwoCohomologyCR[102X( [3XC[103X ) [32X function[133X [33X[0;0YThe first four functions return bases of the corresponding group. The last two functions need to describe a factor of additive abelian groups. They return the following descriptions for these factors.[133X [8X[3Xgcc[103X[8X[108X [33X[0;6Ythe basis of the cocycles of [3XC[103X.[133X [8X[3Xgcb[103X[8X[108X [33X[0;6Ythe basis of the coboundaries of [3XC[103X.[133X [8X[3Xfactor[103X[8X[108X [33X[0;6Ya description of the factor of cocycles by coboundaries. Usually, it would be most convenient to use additive mappings here. However, these are not available in case that [3XA[103X is free abelian and thus we use a description of this additive map as record. This record contains[133X [8X[3Xgens[103X[8X[108X [33X[0;12Ya base for the image.[133X [8X[3Xrels[103X[8X[108X [33X[0;12Yrelative orders for the image.[133X [8X[3Ximgs[103X[8X[108X [33X[0;12Ythe images for the elements in [3Xgcc[103X.[133X [8X[3Xprei[103X[8X[108X [33X[0;12Ypreimages for the elements in [3Xgens[103X.[133X [8X[3Xdenom[103X[8X[108X [33X[0;12Ythe kernel of the map; that is, another basis for [3Xgcb[103X.[133X [33X[0;0YThere is an additional function which can be used to compute the second cohomology group over an arbitrary finitely generated abelian group. The finitely generated abelian group should be realized as a factor of a free abelian group modulo a lattice. The function is called as[133X [1X8.2-2 TwoCohomologyModCR[101X [33X[1;0Y[29X[2XTwoCohomologyModCR[102X( [3XC[103X, [3Xlat[103X ) [32X function[133X [33X[0;0Ywhere [3XC[103X is a cohomology record and [3Xlat[103X is a basis for a sublattice of a free abelian module. The output format is the same as for [10XTwoCohomologyCR[110X.[133X [1X8.3 [33X[0;0YExtended 1-cohomology[133X[101X [33X[0;0YIn some cases more information on the first cohomology group is of interest. In particular, if we have an internal module given and we want to compute the complements using the first cohomology group, then we need additional information. This extended version of first cohomology is obtained by the following functions.[133X [1X8.3-1 OneCoboundariesEX[101X [33X[1;0Y[29X[2XOneCoboundariesEX[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns a record consisting of the entries[133X [8X[3Xbasis[103X[8X[108X [33X[0;6Ya basis for [22XB^1(U, A) ≤ A^n[122X.[133X [8X[3Xtransf[103X[8X[108X [33X[0;6YThere is a derivation mapping from [22XA[122X to [22XB^1(U,A)[122X. This mapping is described here as transformation from [22XA[122X to [3Xbasis[103X.[133X [8X[3Xfixpts[103X[8X[108X [33X[0;6Ythe fixpoints of [22XA[122X. This is also the kernel of the derivation mapping.[133X [1X8.3-2 OneCocyclesEX[101X [33X[1;0Y[29X[2XOneCocyclesEX[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns a record consisting of the entries[133X [8X[3Xbasis[103X[8X[108X [33X[0;6Ya basis for [22XZ^1(U, A) ≤ A^n[122X.[133X [8X[3Xtransl[103X[8X[108X [33X[0;6Ya special solution. This is only of interest in case that [22XC[122X is an internal module and in this case it gives the translation vector in [22XA^n[122X used to obtain complements corresponding to the elements in [22Xbasis[122X. If [22XC[122X is not an internal module, then this vector is always the zero vector.[133X [1X8.3-3 OneCohomologyEX[101X [33X[1;0Y[29X[2XOneCohomologyEX[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns the combined information on the first cohomology group.[133X [1X8.4 [33X[0;0YExtensions and Complements[133X[101X [33X[0;0YThe natural applications of first and second cohomology group is the determination of extensions and complements. Let [22XC[122X be a cohomology record.[133X [1X8.4-1 ComplementCR[101X [33X[1;0Y[29X[2X ComplementCR[102X( [3XC[103X, [3Xc[103X ) [32X function[133X [33X[0;0Yreturns the complement corresponding to the 1-cocycle [3Xc[103X. In the case that [3XC[103X is an external module, we construct the split extension of [22XU[122X with [22XA[122X first and then determine the complement. In the case that [3XC[103X is an internal module, the vector [3Xc[103X must be an element of the affine space corresponding to the complements as described by [10XOneCocyclesEX[110X.[133X [1X8.4-2 ComplementsCR[101X [33X[1;0Y[29X[2X ComplementsCR[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns all complements using the correspondence to [22XZ^1(U,A)[122X. Further, this function returns fail, if [22XZ^1(U,A)[122X is infinite.[133X [1X8.4-3 ComplementClassesCR[101X [33X[1;0Y[29X[2X ComplementClassesCR[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns complement classes using the correspondence to [22XH^1(U,A)[122X. Further, this function returns fail, if [22XH^1(U,A)[122X is infinite.[133X [1X8.4-4 ComplementClassesEfaPcps[101X [33X[1;0Y[29X[2X ComplementClassesEfaPcps[102X( [3XU[103X, [3XN[103X, [3Xpcps[103X ) [32X function[133X [33X[0;0YLet [22XN[122X be a normal subgroup of [22XU[122X. This function returns the complement classes to [22XN[122X in [22XU[122X. The classes are computed by iteration over the [22XU[122X-invariant efa series of [22XN[122X described by [3Xpcps[103X. If at some stage in this iteration infinitely many complements are discovered, then the function returns fail. (Even though there might be only finitely many conjugacy classes of complements to [22XN[122X in [22XU[122X.)[133X [1X8.4-5 ComplementClasses[101X [33X[1;0Y[29X[2X ComplementClasses[102X( [[3XV[103X, ][3XU[103X, [3XN[103X ) [32X function[133X [33X[0;0YLet [22XN[122X and [22XU[122X be normal subgroups of [22XV[122X with [22XN ≤ U ≤ V[122X. This function attempts to compute the [22XV[122X-conjugacy classes of complements to [22XN[122X in [22XU[122X. The algorithm proceeds by iteration over a [22XV[122X-invariant efa series of [22XN[122X. If at some stage in this iteration infinitely many complements are discovered, then the algorithm returns fail.[133X [1X8.4-6 ExtensionCR[101X [33X[1;0Y[29X[2XExtensionCR[102X( [3XC[103X, [3Xc[103X ) [32X function[133X [33X[0;0Yreturns the extension corresponding to the 2-cocycle [22Xc[122X.[133X [1X8.4-7 ExtensionsCR[101X [33X[1;0Y[29X[2XExtensionsCR[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns all extensions using the correspondence to [22XZ^2(U,A)[122X. Further, this function returns fail, if [22XZ^2(U,A)[122X is infinite.[133X [1X8.4-8 ExtensionClassesCR[101X [33X[1;0Y[29X[2XExtensionClassesCR[102X( [3XC[103X ) [32X function[133X [33X[0;0Yreturns extension classes using the correspondence to [22XH^2(U,A)[122X. Further, this function returns fail, if [22XH^2(U,A)[122X is infinite.[133X [1X8.4-9 SplitExtensionPcpGroup[101X [33X[1;0Y[29X[2XSplitExtensionPcpGroup[102X( [3XU[103X, [3Xmats[103X ) [32X function[133X [33X[0;0Yreturns the split extension of [3XU[103X by the [22XU[122X-module described by [3Xmats[103X.[133X [1X8.5 [33X[0;0YConstructing pcp groups as extensions[133X[101X [33X[0;0YThis section contains an example application of the second cohomology group to the construction of pcp groups as extensions. The following constructs extensions of the group of upper unitriangular matrices with its natural lattice.[133X [4X[32X Example [32X[104X [4X[28X# get the group and its matrix action[128X[104X [4X[25Xgap>[125X [27XG := UnitriangularPcpGroup(3,0);[127X[104X [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X [4X[25Xgap>[125X [27Xmats := G!.mats;[127X[104X [4X[28X[ [ [ 1, 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],[128X[104X [4X[28X [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],[128X[104X [4X[28X [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ][128X[104X [4X[28X[128X[104X [4X[28X# set up the cohomology record[128X[104X [4X[25Xgap>[125X [27XC := CRRecordByMats(G,mats);;[127X[104X [4X[28X[128X[104X [4X[28X# compute the second cohomology group[128X[104X [4X[25Xgap>[125X [27Xcc := TwoCohomologyCR(C);;[127X[104X [4X[28X[128X[104X [4X[28X# the abelian invariants of H^2(G,M)[128X[104X [4X[25Xgap>[125X [27Xcc.factor.rels;[127X[104X [4X[28X[ 2, 0, 0 ][128X[104X [4X[28X[128X[104X [4X[28X# construct an extension which corresponds to a cocycle that has[128X[104X [4X[28X# infinite image in H^2(G,M)[128X[104X [4X[25Xgap>[125X [27Xc := cc.factor.prei[2];[127X[104X [4X[28X[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1 ][128X[104X [4X[25Xgap>[125X [27XH := ExtensionCR( CR, c);[127X[104X [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0 ][128X[104X [4X[28X[128X[104X [4X[28X# check that the extension does not split - get the normal subgroup[128X[104X [4X[25Xgap>[125X [27XN := H!.module;[127X[104X [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X [4X[28X[128X[104X [4X[28X# create the interal module[128X[104X [4X[25Xgap>[125X [27XC := CRRecordBySubgroup(H,N);;[127X[104X [4X[28X[128X[104X [4X[28X# use the complements routine[128X[104X [4X[25Xgap>[125X [27XComplementClassesCR(C);[127X[104X [4X[28X[ ][128X[104X [4X[32X[104X
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