  
  [1X1 Introduction[0X
  
  
  [1X1.1 General aims of [5XWedderga[1X package[0X
  
  The  title  ``[5XWedderga[0m''  stands  for  ``[12XWedder[0mburn  decomposition  of [12Xg[0mroup
  [12Xa[0mlgebras''.  This  is  a [5XGAP[0m package to compute the simple components of the
  Wedderburn decomposition of semisimple group algebras. So the main functions
  of  the  package  returns  a  list  of  simple  algebras whose direct sum is
  isomorphic to the group algebra given as input.
  
  The  method implemented by the package produces the Wedderburn decomposition
  of  a group algebra FG provided G is a finite group and F is either a finite
  field  of  characteristic  coprime  to  the order of G, or an abelian number
  field (i.e. a subfield of a finite cyclotomic extension of the rationals).
  
  Other  functions  of  [5XWedderga[0m  compute the primitive central idempotents of
  semisimple group algebras.
  
  The  package  also  provides  functions to construct crossed products over a
  group  with  coefficients  in  an  associative  ring  with  identity and the
  multiplication determined by a given action and twisting.
  
  
  [1X1.2 Main functions of [5XWedderga[1X package[0X
  
  The  main  functions  of  [5XWedderga[0m  are  [2XWedderburnDecomposition[0m ([14X2.1-1[0m) and
  [2XWedderburnDecompositionInfo[0m ([14X2.1-2[0m).
  
  [2XWedderburnDecomposition[0m ([14X2.1-1[0m) computes a list of simple algebras such that
  their  direct product is isomorphic to the group algebra FG, given as input.
  Thus,  the  direct  product  of  the entries of the output is the [13XWedderburn
  decomposition[0m ([14X7.3[0m) of FG.
  
  If  F is an abelian number field then the entries of the output are given as
  matrix  algebras  over  cyclotomic algebras (see [14X7.11[0m), thus, the entries of
  the  output  of  [2XWedderburnDecomposition[0m  ([14X2.1-1[0m)  are  realizations  of the
  [13XWedderburn  components[0m  ([14X7.3[0m)  of FG as algebras which are [13XBrauer equivalent[0m
  ([14X7.5[0m)  to  [13Xcyclotomic  algebras[0m  ([14X7.11[0m). Recall that the Brauer-Witt Theorem
  ensures  that  every  simple  factor of a semisimple group ring FG is Brauer
  equivalent  (that  is  represents  the same class in the Brauer group of its
  centre)  to  a  cyclotomic  algebra  ([Yam74]. In this case the algorithm is
  based  in  a  computational oriented proof of the Brauer-Witt Theorem due to
  Olteanu  [Olt07]  which  uses  previous  work by Olivieri, del Río and Simón
  [ORS04] for rational group algebras of [13Xstrongly monomial groups[0m ([14X7.16[0m).
  
  The Wedderburn components of FG are also matrix algebras over division rings
  which  are  finite  extensions  of  the  field F. If F is finite then by the
  Wedderburn  theorem these division rings are finite fields. In this case the
  output  of  [2XWedderburnDecomposition[0m  ([14X2.1-1[0m) represents the factors of FG as
  matrix algebras over finite extensions of the field F.
  
  In   theory   [5XWedderga[0m  could  handle  the  calculation  of  the  Wedderburn
  decomposition  of group algebras of groups of arbitrary size but in practice
  if  the  order of the group is greater than 5000 then the program may crash.
  The  way  the  group  is  given is relevant for the performance. Usually the
  program works better for groups given as permutation groups or pc groups.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QG := GroupRing( Rationals, SymmetricGroup(4) );[0X
    [4X<algebra-with-one over Rationals, with 2 generators>[0X
    [4Xgap> WedderburnDecomposition(QG);[0X
    [4X[ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ),[0X
    [4X  <crossed product with center Rationals over CF(3) of a group of size 2> ][0X
    [4Xgap> FG := GroupRing( CF(5), SymmetricGroup(4) );[0X
    [4X<algebra-with-one over CF(5), with 2 generators>[0X
    [4Xgap> WedderburnDecomposition( FG );[0X
    [4X[ CF(5), CF(5), ( CF(5)^[ 3, 3 ] ), ( CF(5)^[ 3, 3 ] ),[0X
    [4X  <crossed product with center CF(5) over AsField( CF(5), CF([0X
    [4X    15) ) of a group of size 2> ][0X
    [4Xgap> FG := GroupRing( GF(5), SymmetricGroup(4) ); [0X
    [4X<algebra-with-one over GF(5), with 2 generators>[0X
    [4Xgap> WedderburnDecomposition( FG );[0X
    [4X[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [0X
    [4X  ( GF(5)^[ 3, 3 ] ), ( GF(5)^[ 3, 3 ] ) ][0X
    [4Xgap> FG := GroupRing( GF(5), SmallGroup(24,3) );[0X
    [4X<algebra-with-one over GF(5), with 4 generators>[0X
    [4Xgap> WedderburnDecomposition( FG );[0X
    [4X[ ( GF(5)^[ 1, 1 ] ), ( GF(5^2)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [0X
    [4X  ( GF(5^2)^[ 2, 2 ] ), ( GF(5)^[ 3, 3 ] ) ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  Instead  of  [2XWedderburnDecomposition[0m  ([14X2.1-1[0m),  that  returns  a list of [5XGAP[0m
  objects,   [2XWedderburnDecompositionInfo[0m   ([14X2.1-2[0m)   returns   the   numerical
  description of these objects. See Section [14X7.12[0m for theoretical background.
  
  
  [1X1.3 Installation and system requirements[0X
  
  [5XWedderga[0m  does  not  use  external  binaries  and,  therefore, works without
  restrictions  on the type of the operating system. It is designed for [5XGAP[0m4.4
  and no compatibility with previous releases of [5XGAP[0m4 is guaranteed.
  
  To  use the [5XWedderga[0m online help it is necessary to install the [5XGAP[0m4 package
  [5XGAPDoc[0m  by  Frank Lübeck and Max Neunhöffer, which is available from the [5XGAP[0m
  site or from [7Xhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[0m.
  
  [5XWedderga[0m  is distributed in standard formats ([11Xtar.gz[0m, [11Xtar.bz2[0m, [11X-win.zip[0m) and
  can  be  obtained  from  [7Xhttp://www.um.es/adelrio/wedderga.htm[0m,  its  mirror
  [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/wedderga.htm[0m      or      the     page
  [7Xhttp://www.gap-system.org/Packages/wedderga.html[0m  at  the  [5XGAP[0m web site. The
  latter also offers [11Xzoo[0m-archive. To unpack the archive [11Xwedderga-4.3.2.zoo[0m you
  need  the  program  [11Xunzoo[0m,  which  can  be  obtained  from  the [5XGAP[0m homepage
  [7Xhttp://www.gap-system.org/[0m   (see   section   `Distribution').   To  install
  [5XWedderga[0m,  copy  this  archive  into  the  [11Xpkg[0m  subdirectory  of your [5XGAP[0m4.4
  installation. The subdirectory [11Xwedderga[0m will be created in the [11Xpkg[0m directory
  after the following command:
  
  [10Xunzoo -x wedderga-4.3.2.zoo[0m
  
  When  you  don't have access to the directory of your main [5XGAP[0m installation,
  you can also install the package [13Xoutside the [5XGAP[0m[13X main directory[0m by unpacking
  it  inside  a  directory  [11XMYGAPDIR/pkg[0m. Then to be able to load Wedderga you
  need to call GAP with the [10X-l ";MYGAPDIR"[0m option.
  
  Installation using other archive formats is performed in a similar way.
  
