  
  [1X3 [33X[0;0YImplementation Details[133X[101X
  
  [33X[0;0YIn  this  chapter we describe the approach used to store the normalized unit
  group  of  the  group  algebra  in the library, and to reconstruct the group
  [22XV(KG)[122X from the stored information.[133X
  
  
  [1X3.1 [33X[0;0YSaving the data[133X[101X
  
  [33X[0;0YTo  compute  the pc-presentation of the normalized unit group of the modular
  group algebra of a finite [22Xp[122X-group we used the function [10XPcNormalizedUnitGroup[110X
  from  the [5XLAGUNA[105X package. It uses the algorithm descibed in [Bov98]. See the
  [5XLAGUNA[105X manual [BKRS] for more details.[133X
  
  [33X[0;0YWhen  this  group  is  computed,  the  main  idea  is  to  use  [5XGAP[105X function
  [10XCodePcGroup[110X  that returns the code for the polycyclic generating sequence of
  the  group,  and  then  to  create  the  group  from this code using the [5XGAP[105X
  function [10XPcGroupCode[110X.[133X
  
  [33X[0;0YThe  resulting  code  could be very long, and to compress it we used the [5XGAP[105X
  function  [10XHexStringInt[110X  than  returns a string that represents the code with
  hexa-decimal  digits  (using  A-F  as digits 10-15). The inverse translation
  then  can  be  performed with the [5XGAP[105X function [10XIntHexString[110X. This allowed to
  save  almost 20 MB of space on groups of order 128 and reduce the total size
  of their database to 90 MB.[133X
  
  [33X[0;0YFor  groups  of  order  128 we decided to compress the library with the [11Xgzip[111X
  program,  and  then uncompress each file "on fly" when it is requested. This
  allowed us to reduce the size of their part in the library from 90 to 12 MB,
  which  is already quite reasonable. Of course, there is some little overhead
  arising  from the uncompression and subsequent translation from hexa-decimal
  notation,  but  it  is  neglectible comparatively with the total time of the
  computation of [22XV(KG)[122X from scratch.[133X
  
  [33X[0;0YThere  is  one more thing that needs to be stored together with this code to
  make  it  sure that we will correctly identify the underlying group [22XG[122X of the
  group  algebra  [22XKG[122X  with  its image in the pc-presentation of the normalized
  unit group [22XV(KG)[122X.[133X
  
  [33X[0;0YThe  group [22XG[122X is extracted from the [5XGAP[105X Small Groups Library, so it is always
  the same, unless its description in the library will be changed (and it will
  be  an  important  task  of  [5XUnitLib[105X maintaner to update the package in this
  case!), and here we are safe from inconsistencies.[133X
  
  [33X[0;0YBut  the  next stage is the computation of generators of the normalized unit
  group  [22XV(KG)[122X, and the first step is the dimension basis of the group [22XG[122X, that
  can  be  computed  using  the  [5XLAGUNA[105X  function [10XDimensionBasis[110X. To avoid the
  influence  of  possible  changes in [5XGAP[105X or usage of random methods, we store
  (in  compacted  form)  the information about the dimension basis of [22XG[122X in the
  [5XUnitLib[105X.[133X
  
  [33X[0;0YAll  further procedures are implemented inside the [5XLAGUNA[105X package, and their
  result is uniquely determined and predictable.[133X
  
  [33X[0;0YFor most groups all information is stored in a single file. However, this is
  not  the case for groups of order 243, where we have about 30 MB of data for
  67  groups.  For  these  groups  we  provide  a  solution  on  the  base  of
  Web-services.  The  information about the dimension basis is stored locally,
  while  the  codes for polycyclic generating sequences are available from the
  [5XUnitLib[105X  homepage,  and  the  package  will  access  them using the function
  [2XSingleHTTPRequest[102X ([14XIO: SingleHTTPRequest[114X) from the [5XIO[105X package.[133X
  
  [33X[0;0YFor  the  reader  interested  in more details, the package contains the file
  [11Xunitlib/lib/genlib.g[111X  with the function [10XCreatePcNormalizedUnitGroupsLibrary[110X,
  that  creates library files for groups of a given prime power order, and the
  file          [11Xunitlib/lib/genpar.g[111X         with         the         function
  [10XParCreatePcNormalizedUnitGroupsLibrary[110X, which is the parallel version of the
  previous function and must be used with the [5XParGAP[105X package [Coo].[133X
  
  
  [1X3.2 [33X[0;0YReading the data[133X[101X
  
  [33X[0;0YTo  reconstruct  the  normalized  unit group [22XV(KG)[122X from the library, we need
  only to know the catalogue number of the underlying group [22XG[122X in the [5XGAP[105X Small
  Groups Library.[133X
  
  [33X[0;0YWe  use  the  same  numbering  as in the [5XGAP[105X Small Group Library, so [5XUnitLib[105X
  finds  the  appropriate  library  file(s) and reads from it the code for the
  polycyclic  generating  sequence  of  [22XV(KG)[122X  and  the  information about the
  dimension basis of [22XG[122X used for the computation of this code.[133X
  
  [33X[0;0YThen  [22XV(KG)[122X  is  created  from  the  code using the [5XGAP[105X function [2XPcGroupCode[102X
  ([14XReference:  PcGroupCode[114X).  We  also  create  [22XG[122X  using  the [5XGAP[105X Small Groups
  Library.[133X
  
  [33X[0;0YNow  to [21Xglue[121X the group [22XV(KG)[122X with the underlying group [22XG[122X properly, the value
  of  the  attribute  [2XDimensionBasis[102X  ([14XLAGUNA:  DimensionBasis[114X) of [22XG[122X is set in
  accordance with the data retrieved from the library. This will guarantee the
  correct  construction  of  [2XNaturalBijectionToPcNormalizedUnitGroup[102X  ([14XLAGUNA:
  NaturalBijectionToPcNormalizedUnitGroup[114X)                                 and
  [2XNaturalBijectionToNormalizedUnitGroup[102X                               ([14XLAGUNA:
  NaturalBijectionToNormalizedUnitGroup[114X) by the [5XLAGUNA[105X package.[133X
  
  [33X[0;0YIt  remains  now  to make only several technical steps, such as constructing
  the  group  algebra  [22XKG[122X  over the appropriate field [22XK[122X, and storing [22XKG[122X in the
  attribute  [2XUnderlyingGroupRing[102X  ([14XLAGUNA:  UnderlyingGroupRing[114X)  of [22XV(KG)[122X and
  [22XV(KG)[122X in the attribute [2XPcNormalizedUnitGroup[102X ([14XLAGUNA: PcNormalizedUnitGroup[114X)
  of [22XKG[122X.[133X
  
