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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

Path: gap4r8 / pkg / aclib / doc / manual.dvi
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��x;����:�;��;��La�A;��Lb�A;��Lc�A;��Ld�E�)�H�of�Hthose�gener���-���ators���which���are�present�in�the�dening�generating�set�of�the�matrix�group.�This�ne��w�ordering�of�the�generators�is�then���used��to�dene�a�polyc�٠yclic�presentation�of�the�gi��v�en�almost�crystallographic�group.�����w������p���YƢ�������M4����P��?��NExample��,����acomputations����8with��.almost������!cr�\%ystallographic����Z�qgr��G�oups����>�ʍ�P4.1��Example��&computations�I��N8��Using���the�functions�a���v��ailable�for�pcp�groups�in�the�share�package����Qpolycyclic��it�is�no�w�easy�to�redo�some�of�the���calculations�w�of�w�[Dek96].�As�a�rst�e�٠xample�we�check�whether�the�groups�indicated�as�torsion�free�in�[Dek96]�are�also���determined���as�torsion���free�ones�by��QGAP�.�In�[Dek96]�these�almost�Bieberbach�groups�are�listed�as�\��37AB-groups".�So���for�J�type�J�\013"�these�are�the�groups�with�parameters��(�Lk�+�A;����0�A;��1�A;��0�A;��1�A;��0)�J��where��Lk�v�is�an�e��v�٠en�inte�ger��s8.�J�Let'�s�look�J�at�some���e�٠xamples��in��QGAP�:����9��gap>�?�G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�true����9�gap>�?�G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�false����Further���,��there�is�also�some�cohomology�information�in�the�tables�of�[Dek96].�In�f��gact,�the�groups�in�this�library�were���obtained��as�e�٠xtensions��LE����of�the�form��N8����Ɵ�1���D!��Z��D!��LE���D!��LQ��D!��1��N8��where,��in�the��4-dimensional�case��LQ����=��LE�:��A=�Dh�Ld�E�Di�.��The�cohomology�information�for�the�particular�e�٠xample�abo�v�e�sho��ws���that��the��groups�determined�by�a�parameter�set��(�Lk����1���A;����Lk����2���A;��Lk����3���A;��Lk����4���A;��Lk����4���A;��Lk����6���)���are��equi��v�alent�as��e�٠xtensions�to�the�groups�deter���-���mined��by�the�parameters��(�Lk����1���A;����Lk����2�����mod��U�2�A;��Lk����3�����mod��2�A;��Lk����4�����mod��2�A;��Lk����5�����mod��2�A;��0)�.��This�is�also�visible�in�nding�torsion:�����
}������LSection��3.�Example�computations�III�80��13���p������9��gap>�?�G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�false����9�gap>�?�G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�true����P4.2��Example��&computations�II��N8��The�"�computation�"�of�cohomology�groups�played�an�important�role�in�the�classication�of�the�almost�Bieberbach�groups���in��[Dek96].��Using��QGAP�,�it�is�no��w�possible�to�check�these�computations.�As�an�e�٠xample�we�consider�the�4-dimensional���almost��Rcrystallographic��Qgroups�of�type�85�on�page�202�of�[Dek96].�This�group��LE�10�has�6�generators.�In�the�table,�one���also��nds�the�information��N8����]��LH��s0�����2��s0�(�LQ�A;����Z�)��=��Z�8��D��(�Z����2���)������2��8��D��Z����4���N8��for���LQ����=����LE�:��A=�Dh�Ld�E�Di���as�abo�٠v�e.��Moreo�v�er���,�the��LQ�{module���Z��is�in�f��gact�the�group��Dh�Ld�E�Di�,�where�the��LQ�-action�comes�from���conjug��7ation��inside���LE�:��.�In�the�case�of�groups�of�type�85,��Z��is�a�tri��vial��LQ�-module.�The�follo��wing�e�٠xample�demonstrates���ho��w��to�(re)compute�this�tw��go-cohomology�group��LH��s0��^��2��s0�(�LQ�A;����Z�)�.����9��gap>�?�G:=AlmostCrystallographicPcpGroup(4,�"085",�false);����9�Pcp�?�group�with�orders�[�2,�4,�0,�0,�0,�0�]����9�gap>�?�GroupGeneratedByd:=Subgroup(G,�[G.6]�);����9�Pcp�?�group�with�orders�[�0�]����9�gap>�?�Q:=G/GroupGeneratedByd;����9�Pcp�?�group�with�orders�[�2,�4,�0,�0,�0�]����9�gap>�?�action:=List(�Pcp(Q),�x�->�[[1]]�);����9�[�?�[�[�1�]�],�[�[�1�]�],�[�[�1�]�],�[�[�1�]�],�[�[�1�]�]�]����9�gap>�?�C:=CRRecordByMats(�Q,�action);;����9�gap>�?�TwoCohomologyCR(�C�).factor.rels;����9�[�?�2,�2,�4,�0�]����This�M�last�line�gi��v�٠es�M�us�the�abelian�in���v�ariants�of�the�second�M�cohomology�group��LH��s0��^��2��s0�(�LQ�A;����Z�)�.�So�we�should�read�this�line�as��N8�����c�LH��s0�����2��s0�(�LQ�A;����Z�)��=��Z����2��8��D�8��Z����2���D��Z����4���D��Z��N8��which��indeed�coincides�with�the�information�in�[Dek96].��N8��P4.3��Example��&computations�III����As���another�application���of�the�capabilities�of�the�combination�of��aclib��and��Qpolycyclic��we�check�some�computations���of��[DM01].��N8�Section�ay5�of�azthe�paper�[DM01]�is�completely�de��v���oted�to�an�e�٠xample�of�the�computation�of�the��LP�-localization�of�a���virtually��Cnilpotent�group,�where��D�LP��is�a�set�of�primes.�Although�it�is�not�our�intention�to�de��v�٠elop�the�theory�of��LP�-���localization��of�groups�at�this�place,�let�us�summarize�some�of�the�main�results�concerning�this�topic�here.���F�٠or���a���set�of�primes��LP�,�we�say�that��Ln�JD�D2�JE�LP��if�and�only�if��Ln��is�a�product�of�primes�in��LP�.�A���group��LG��is�said�to�be��LP�-local���if�Xand�Yonly�if�the�map��A����Tn�����:����LG��D!��LG��:��Lg��D7!��Lg���^��Tn��Y�is�Xbijecti��v�٠e�for�all��Ln����D2��LP���^��E0��l��,�where�X�LP���^��E0���$�is�the�set�of�all�primes�not�in��LP�.���The�Z#�LP�-localization�Z"of�a�group��LG�,�is�a��LP�-local�group��LG����TP��!�together�with�a�morphism��A�/��:�[�LG�[
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����3���(�LK��)��is�nite�in�this�e�٠xample,�we���e�٠xactly�kno��w�what�this��LP�-localization�is.�Let�us�no��w�sho�w��Y�,�ho�w�GAP��can�be�used�to�compute�this��LP�-localization�in���tw��go��cases:����First��case:�The�parameters�are��(�Lk����1���A;����Lk����2���A;��Lk����3���A;��Lk����4���)��=�(2�A;��0�A;��0�A;��0)�����9��gap>�?�G�:=�AlmostCrystallographicPcpGroup(3,�17,�[2,0,0,0]�);����9�Pcp�?�group�with�orders�[�2,�6,�0,�0,�0�]����9�gap>�?�projection�:=�NaturalHomomorphismOnHolonomyGroup(�G�);����9�[�?�g1,�g2,�g3,�g4,�g5�]�->�[�g1,�g2,�identity,�identity,�identity�]����9�gap>�?�F�:=�HolonomyGroup(�G�);����9�Pcp�?�group�with�orders�[�2,�6�]����9�gap>�?�IPprimeD6�:=�Subgroup(�F�,�[F.2^2]�);����9�Pcp�?�group�with�orders�[�3�]����9�gap>�?�K�:=�PreImage(�projection,�IPprimeD6�);����9�Pcp�?�group�with�orders�[�3,�0,�0,�0�]����9�gap>�?�PrintPcpPresentation(�K�);����9�pcp�?�presentation�on�generators�[�g2^2,�g3,�g4,�g5�]����9�g2^2�?�^�3�=�identity����9�g3�?�^�g2^2�=�g3^-1*g4^-1����9�g3�?�^�g2^2^-1�=�g4*g5^-2����9�g4�?�^�g2^2�=�g3*g5^2����9�g4�?�^�g2^2^-1�=�g3^-1*g4^-1*g5^2����9�g4�?�^�g3�=�g4*g5^2����9�g4�?�^�g3^-1�=�g4*g5^-2����9�gap>�?�Gamma3K�:=�CommutatorSubgroup(�K,�CommutatorSubgroup(�K,�K�));����9�Pcp�?�group�with�orders�[�0,�0,�0�]����9�gap>�?�quotient�:=�G/Gamma3K;����9�Pcp�?�group�with�orders�[�2,�6,�3,�3,�2�]����9�gap>�?�S�:=�SylowSubgroup(�quotient,�3);����9�Pcp�?�group�with�orders�[�3,�3,�3�]����9�gap>�?�N�:=�NormalClosure(�quotient,�S);����9�Pcp�?�group�with�orders�[�3,�3,�3�]����9�gap>�?�localization�:=�quotient/N;������썟���LSection��3.�Example�computations�III�80��15���p������9��Pcp�?�group�with�orders�[�2,�2,�2�]����9�gap>�?�PrintPcpPresentation(�localization�);����9�pcp�?�presentation�on�generators�[�g1,�g2,�g3�]����9�g1�?�^�2�=�identity����9�g2�?�^�2�=�identity����9�g3�?�^�2�=�identity����This��sho��ws�that��LG����TP�����W�����D���(���=�����1�Z���^���3��d�2����.��N8�Second��case:�The�parameters�are��(�Lk����1���A;����Lk����2���A;��Lk����3���A;��Lk����4���)��=�(2�A;��0�A;��0�A;��1)����9��gap>�?�G�:=�AlmostCrystallographicPcpGroup(3,�17,�[2,0,0,1]);;����9�gap>�?�projection�:=�NaturalHomomorphismOnHolonomyGroup(�G�);;����9�gap>�?�F�:=�HolonomyGroup(�G�);;����9�gap>�?�IPprimeD6�:=�Subgroup(�F�,�[F.2^2]�);;����9�gap>�?�K�:=�PreImage(�projection,�IPprimeD6�);;����9�gap>�?�Gamma3K�:=�CommutatorSubgroup(�K,�CommutatorSubgroup(�K,�K�));;����9�gap>�?�quotient�:=�G/Gamma3K;;����9�gap>�?�S�:=�SylowSubgroup(�quotient,�3);;����9�gap>�?�N�:=�NormalClosure(�quotient,�S);;����9�gap>�?�localization�:=�quotient/N;����9�Pcp�?�group�with�orders�[�2,�2,�2�]����9�gap>�?�PrintPcpPresentation(�localization�);����9�pcp�?�presentation�on�generators�[�g1,�g2,�g3�]����9�g1�?�^�2�=�identity����9�g2�?�^�2�=�g3����9�g3�?�^�2�=�identity����9�g2�?�^�g1�=�g2*g3����9�g2�?�^�g1^-1�=�g2*g3����In��this�case,�we�see�that��LG����TP����=���DD����4���.���The��reader�can��check�that�these�results�coincide�with�those�obtained�in�[DM01].�Note�also�that�we�used�a�some��what���dierent�scheme�to�compute�this�localization�than�the�one�used�in�[DM01].�W��37e�in���vite�the�reader�to�check�the�same���computations,��tracing�e�٠xactly�the�steps�made�in�[DM01].������Y�����p����C�������NBib���liograph��G�y����<!�����:��[Aus60]��$Louis��CAuslander��s8.��BBieberbach'�s�theorem��Bon�space�groups�and�discrete�uniform�subgroups�of�lie�groups.��KAnn.����$of��Math�,�71(3):579{590,�1960.�������W[Bro82]��$K��enneth��*S.�Bro�wn.��KCohomology��)of�Groups�,�v���olume�87�of��KGrad.�T��L�e�٠xts�in�Math.��Springer���-V����erlag,�Ne��w�Y���gork,����$1982.��������[DE01a]��$Karel��oDekimpe��nand�Bettina�Eick.�Computational�aspects�of�group�e�٠xtensions�and�their�application�in����$topology��Y�.���KSubmitted�,�2001.�������[DE01b]��$Karel��Dekimpe�and�Bettina�Eick.�Computations�with�almost�crystallographic�groups.��KSubmitted�,�2001.������� 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