  
  [1m[4m[31m1. Introduction[0m
  
  
  [1m[4m[31m1.1 General aims of [1mWedderga[1m[4m[31m package[0m
  
  The  title  ``[1mWedderga[0m''  stands  for  ``[1m[46mWedder[0mburn  decomposition  of [1m[46mg[0mroup
  [1m[46ma[0mlgebras''.  This  is  a [1mGAP[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 methods 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 with the order of G, or an abelian number
  field (i.e. a subfield of a finite cyclotomic extension of the rationals).
  
  Other  functions  of  [1mWedderga[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.
  
  
  [1m[4m[31m1.2 Main functions of [1mWedderga[1m[4m[31m package[0m
  
  The  main  functions  of  [1mWedderga[0m  are  [1m[34mWedderburnDecomposition[0m ([1m2.1-1[0m) and
  [1m[34mWedderburnDecompositionInfo[0m ([1m2.1-2[0m).
  
  [1m[34mWedderburnDecomposition[0m ([1m2.1-1[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 [22m[36mWedderburn
  decomposition[0m ([1m7.3[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 [1m7.11[0m), thus, the entries of
  the  output  of  [1m[34mWedderburnDecomposition[0m  ([1m2.1-1[0m)  are  realizations  of the
  [22m[36mWedderburn  components[0m  ([1m7.3[0m)  of FG as algebras which are [22m[36mBrauer equivalent[0m
  ([1m7.5[0m)  to  [22m[36mcyclotomic  algebras[0m  ([1m7.11[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 ([Y74], [O07]).
  
  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  [1m[34mWedderburnDecomposition[0m  ([1m2.1-1[0m) represents the factors of FG as
  matrix algebras over finite extensions of the field F.
  
  .
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> QG := GroupRing( Rationals, SymmetricGroup(4) );[0m
    [22m[35m<algebra-with-one over Rationals, with 2 generators>[0m
    [22m[35mgap> WedderburnDecomposition(QG);[0m
    [22m[35m[ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ),[0m
    [22m[35m  <crossed product with center Rationals over CF(3) of a group of size 2> ][0m
    [22m[35mgap> FG := GroupRing( CF(5), SymmetricGroup(4) );[0m
    [22m[35m<algebra-with-one over CF(5), with 2 generators>[0m
    [22m[35mgap> WedderburnDecomposition( FG );[0m
    [22m[35m[ CF(5), CF(5), ( CF(5)^[ 3, 3 ] ), ( CF(5)^[ 3, 3 ] ),[0m
    [22m[35m  <crossed product with center CF(5) over AsField( CF(5), CF([0m
    [22m[35m    15) ) of a group of size 2> ][0m
    [22m[35mgap> FG := GroupRing( GF(5), SymmetricGroup(4) ); [0m
    [22m[35m<algebra-with-one over GF(5), with 2 generators>[0m
    [22m[35mgap> WedderburnDecomposition( FG );[0m
    [22m[35m[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [0m
    [22m[35m  ( GF(5)^[ 3, 3 ] ), ( GF(5)^[ 3, 3 ] ) ][0m
    [22m[35mgap> FG := GroupRing( GF(5), SmallGroup(24,3) );[0m
    [22m[35m<algebra-with-one over GF(5), with 4 generators>[0m
    [22m[35mgap> WedderburnDecomposition( FG );[0m
    [22m[35m[ ( GF(5)^[ 1, 1 ] ), ( GF(5^2)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [0m
    [22m[35m  ( GF(5^2)^[ 2, 2 ] ), ( GF(5)^[ 3, 3 ] ) ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  Instead  of  [1m[34mWedderburnDecomposition[0m  ([1m2.1-1[0m),  that  returns  a list of [1mGAP[0m
  objects,   [1m[34mWedderburnDecompositionInfo[0m   ([1m2.1-2[0m)   returns   the   numerical
  description of these objects. See Section [1m7.12[0m for theoretical background.
  
  
  [1m[4m[31m1.3 Installation and system requirements[0m
  
  [1mWedderga[0m  does  not  use  external  binaries  and,  therefore, works without
  restrictions  on the type of the operating system. It is designed for [1mGAP[0m4.4
  and no compatibility with previous releases of [1mGAP[0m4 is guaranteed.
  
  To  use the [1mWedderga[0m online help it is necessary to install the [1mGAP[0m4 package
  [1mGAPDoc[0m  by  Frank L\"ubeck and Max Neunh\"offer, which is available from the
  [1mGAP[0m site or from [34mhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[0m.
  
  [1mWedderga[0m is distributed in standard formats ([1mzoo[0m, [1mtar.gz[0m, [1mtar.bz2[0m, [1m-win.zip[0m)
  and  can  be  obtained from [34mhttp://www.um.es/adelrio/wedderga.htm[0m. To unpack
  the  archive  [1mwedderga-4.2.zoo[0m  you  need  the  program  [1munzoo[0m, which can be
  obtained  from  the  [1mGAP[0m  homepage  [34mhttp://www.gap-system.org/[0m  (see section
  `Distribution').  To  install  [1mWedderga[0m,  copy  this  archive  into  the [1mpkg[0m
  subdirectory  of your [1mGAP[0m4.4 installation. The subdirectory [1mwedderga[0m will be
  created in the [1mpkg[0m directory after the following command:
  
  [22m[32munzoo -x wedderga-4.2.zoo[0m
  
