  
  [1X2 Wedderburn decomposition[0X
  
  
  [1X2.1 Wedderburn decomposition[0X
  
  [1X2.1-1 WedderburnDecomposition[0m
  
  [2X> WedderburnDecomposition( [0X[3XFG[0X[2X ) ___________________________________[0Xattribute
  [6XReturns:[0X  A list of simple algebras.
  
  The input [3XFG[0m should be a group algebra of a finite group G over the field F,
  where  F  is  either  an  abelian  number field (i.e. a subfield of a finite
  cyclotomic  extension  of the rationals) or a finite field of characteristic
  coprime with the order of G.
  
  The  function  returns  the  list  of all [13XWedderburn components[0m ([14X7.3[0m) of the
  group  algebra  [3XFG[0m.  If  F  is  an abelian number field then each Wedderburn
  component  is given as a matrix algebra of a [13Xcyclotomic algebra[0m ([14X7.11[0m). If F
  is  a  finite  field  then  the  Wedderburn  components  are given as matrix
  algebras over finite fields.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> WedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) );[0X
    [4X[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ),[0X
    [4X  ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ][0X
    [4Xgap> WedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) );[0X
    [4X[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[0X
    [4X  <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[0X
    [4X    [ 1, 7 ]), CF(8) ) of a group of size 2> ][0X
    [4Xgap> WedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) );[0X
    [4X[ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ),[0X
    [4X  <crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40,[0X
    [4X    [ 1, 31 ]), CF(40) ) of a group of size 2> ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  The  previous examples show that if D_16 denotes the dihedral group of order
  16  then the [13XWedderburn decomposition[0m ([14X7.3[0m) of F_5 D_16, ℚ D_16 and ℚ (xi_5)
  D_16 are respectively
  
  
       \mathbb F_5 D_{16} = 4 \mathbb F_5 \oplus M_2( \mathbb F_5 )
       \oplus M_2( \mathbb F_{25} ),
  
  
  
       ℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus (K(\xi_8)/K,t),
  
  
  and
  
  
       ℚ (\xi_5) D_{16} = 4 ℚ (\xi_5) \oplus M_2( ℚ (\xi_5) ) \oplus
       (F(\xi_{40})/F,t),
  
  
  where  (K(xi_8)/K,t) is a [13Xcyclotomic algebra[0m ([14X7.11[0m) with the centre K=NF(8,[
  1,  7 ])= ℚ (sqrt2), (F(xi_40)/F,t) = ℚ (sqrt2,xi_5) is a cyclotomic algebra
  with centre F=NF(40,[ 1, 31 ]) and xi_n denotes a n-th root of unity.
  
  Two more examples:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );[0X
    [4X[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[0X
    [4X  <crossed product with center Rationals over CF(3) of a group of size 2>,[0X
    [4X  ( CF(3)^[ 2, 2 ] ), <crossed product with center Rationals over CF([0X
    [4X    3) of a group of size 2>, <crossed product with center NF(8,[0X
    [4X    [ 1, 7 ]) over AsField( NF(8,[ 1, 7 ]), CF(8) ) of a group of size 2>,[0X
    [4X  <crossed product with center Rationals over CF(12) of a group of size 4> ][0X
    [4Xgap> WedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) );[0X
    [4X[ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ),[0X
    [4X  ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ),[0X
    [4X  <crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24,[0X
    [4X    [ 1, 7 ]), CF(24) ) of a group of size 2>,[0X
    [4X  ( <crossed product with center CF(3) over AsField( CF(3), CF([0X
    [4X    12) ) of a group of size 2>^[ 2, 2 ] ) ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  In  some  cases,  in  characteristic  zero,  some  entries  of the output of
  [2XWedderburnDecomposition[0m   do   not  provide  full  matrix  algebras  over  a
  [13Xcyclotomic  algebra[0m  ([14X7.11[0m), but "fractional matrix algebras". That entry is
  not an algebra that can be used as a [5XGAP[0m object. Instead it is a pair formed
  by  a  rational giving the "size" of the matrices and a crossed product. See
  [14X7.3[0m for a theoretical explanation of this phenomenon. In this case a warning
  message is displayed.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QG:=GroupRing(Rationals,SmallGroup(240,89));[0X
    [4X<algebra-with-one over Rationals, with 2 generators>[0X
    [4Xgap> WedderburnDecomposition(QG);[0X
    [4XWedderga: Warning!!![0X
    [4XSome of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!![0X
    [4X[0X
    [4X[ Rationals, Rationals, <crossed product with center Rationals over CF([0X
    [4X    5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ),[0X
    [4X  ( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ),[0X
    [4X  <crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12,[0X
    [4X    [ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>,[0X
    [4X  [ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[0X
    [4X        [ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ]  [0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.1-2 WedderburnDecompositionInfo[0m
  
  [2X> WedderburnDecompositionInfo( [0X[3XFG[0X[2X ) _______________________________[0Xattribute
  [6XReturns:[0X  A  list  with  each  entry a numerical description of a [13Xcyclotomic
            algebra[0m ([14X7.11[0m).
  
  The input [3XFG[0m should be a group algebra of a finite group G over the field F,
  where  F  is  either  an  abelian  number field (i.e. a subfield of a finite
  cyclotomic  extension  of the rationals) or a finite field of characteristic
  coprime to the order of G.
  
  This function is a numerical counterpart of [2XWedderburnDecomposition[0m ([14X2.1-1[0m).
  
  It returns a list formed by lists of lengths 2, 4 or 5.
  
  The lists of length 2 are of the form
  
  
       [n,F],
  
  
  where  n  is  a  positive  integer  and F is a field. It represents the nx n
  matrix algebra M_n(F) over the field F.
  
  The lists of length 4 are of the form
  
  
       [n,F,k,[d,\alpha,\beta]],
  
  
  where   F  is  a  field  and  n,k,d,alpha,beta  are  non-negative  integers,
  satisfying  the conditions mentioned in Section [14X7.12[0m. It represents the nx n
  matrix algebra M_n(A) over the cyclic algebra
  
  
       A=F(\xi_k)[u | \xi_k^u = \xi_k^{\alpha}, u^d = \xi_k^{\beta}],
  
  
  where xi_k is a primitive k-th root of unity.
  
  The lists of length 5 are of the form
  
  
       [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j
       \le m} ],
  
  
  where  F  is  a  field and n,k,d_i,alpha_i,beta_i,gamma_i,j are non-negative
  integers.  It  represents the nx n matrix algebra M_n(A) over the [13Xcyclotomic
  algebra[0m ([14X7.11[0m)
  
  
       A = F(\xi_k)[g_1,\ldots,g_m \mid \xi_k^{g_i} = \xi_k^{\alpha_i},
       g_i^{d_i}=\xi_k^{\beta_i}, g_jg_i=\xi_k^{\gamma_{ij}} g_i g_j],
  
  
  where xi_k is a primitive k-th root of unity (see [14X7.12[0m).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> WedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) );[0X
    [4X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],[0X
    [4X  [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ] ][0X
    [4Xgap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) );[0X
    [4X[ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ],[0X
    [4X  [ 1, NF(40,[ 1, 31 ]), 8, [ 2, 7, 0 ] ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  The  interpretation  of  the  previous  example  gives rise to the following
  [13XWedderburn  decompositions[0m  ([14X7.3[0m), where D_16 is the dihedral group of order
  16 and xi_5 is a primitive 5-th root of unity.
  
  
       ℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus M_2( ℚ (\sqrt{2})).
  
  
  
       ℚ (\xi_5) D_{16} = 4 ℚ (\xi_5) \oplus M_2( ℚ (\xi_5)) \oplus M_2(
       ℚ (\xi_5,\sqrt{2})).
  
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;;rel:=[a^8,a^4*b^2,b^-1*a*b*a];;[0X
    [4Xgap> Q16:=F/rel;; QQ16:=GroupRing( Rationals, Q16 );;[0X
    [4Xgap> QS4:=GroupRing( Rationals, SymmetricGroup(4) );;[0X
    [4Xgap> WedderburnDecomposition(QQ16);[0X
    [4X[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[0X
    [4X  <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[0X
    [4X    [ 1, 7 ]), CF(8) ) of a group of size 2> ][0X
    [4Xgap> WedderburnDecomposition( QS4 );[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> WedderburnDecompositionInfo(QQ16);[0X
    [4X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [0X
    [4X  [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ] ][0X
    [4Xgap> WedderburnDecompositionInfo(QS4);  [0X
    [4X[ [ 1, Rationals ], [ 1, Rationals ], [ 3, Rationals ], [ 3, Rationals ], [0X
    [4X  [ 1, Rationals, 3, [ 2, 2, 0 ] ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  In  the  previous  example  we  computed the Wedderburn decomposition of the
  rational  group  algebra  ℚ Q_16 of the quaternion group of order 16 and the
  rational group algebra ℚ S_4 of the symmetric group on four letters. For the
  two   group  algebras  we  used  both  [2XWedderburnDecomposition[0m  ([14X2.1-1[0m)  and
  [2XWedderburnDecompositionInfo[0m.
  
  The output of [2XWedderburnDecomposition[0m ([14X2.1-1[0m) shows that
  
  
       ℚ Q_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus A,
  
  
  
       ℚ S_{4} = 2 ℚ \oplus 2 M_3( ℚ ) \oplus B,
  
  
  where A and B are [13Xcrossed products[0m ([14X7.6[0m) with coefficients in the cyclotomic
  fields  ℚ (xi_8) and ℚ (xi_3) respectively. This output can be used as a [5XGAP[0m
  object,  but  it  does  not  give  clear information on the structure of the
  algebras A and B.
  
  The  numerical  information  displayed  by [2XWedderburnDecompositionInfo[0m means
  that
  
  
       A = ℚ (\xi|\xi^8=1)[g | \xi^g = \xi^7 = \xi^{-1}, g^2 = \xi^4 =
       -1],
  
  
  
       B = ℚ (\xi|\xi^3=1)[g | \xi^g = \xi^2 = \xi^{-1}, g^2 = 1].
  
  
  Both  A  and B are quaternion algebras over its centre which is ℚ (xi+xi^-1)
  and the former is equal to ℚ (sqrt2) and ℚ respectively.
  
  In  B,  one  has  (g+1)(g-1)=0, while g is neither 1 nor -1. This shows that
  B=M_2( ℚ ). However the relation g^2=-1 in A shows that
  
  
       A=ℚ (\sqrt{2})[i,g|i^2=g^2=-1,ig=-gi]
  
  
  and  so A is a division algebra with centre ℚ (sqrt2), which is a subalgebra
  of  the algebra of Hamiltonian quaternions. This could be deduced also using
  well known methods on cyclic algebras (see e.g. [Rei03]).
  
  The next example shows the output of [10XWedderburnDecompositionInfo[0m for ℚ G and
  ℚ  (xi_3)  G,  where  G=SmallGroup(48,15).  The user can compare it with the
  output of [2XWedderburnDecomposition[0m ([14X2.1-1[0m) for the same group in the previous
  section. Notice that the last entry of the [13XWedderburn decomposition[0m ([14X7.3[0m) of
  ℚ  G  is  not  given  as  a matrix algebra of a cyclic algebra. However, the
  corresponding entry of ℚ (xi_3) G is a matrix algebra of a cyclic algebra.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> WedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) );[0X
    [4X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],[0X
    [4X  [ 2, Rationals ], [ 1, Rationals, 3, [ 2, 2, 0 ] ], [ 2, CF(3) ],[0X
    [4X  [ 1, Rationals, 6, [ 2, 5, 0 ] ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ],[0X
    [4X  [ 1, Rationals, 12, [ [ 2, 5, 9 ], [ 2, 7, 0 ] ], [ [ 9 ] ] ] ][0X
    [4Xgap> WedderburnDecompositionInfo( GroupRing( CF(3), SmallGroup(48,15) ) );[0X
    [4X[ [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 2, CF(3) ],[0X
    [4X  [ 2, CF(3), 3, [ 1, 1, 0 ] ], [ 2, CF(3) ], [ 2, CF(3) ],[0X
    [4X  [ 2, CF(3), 6, [ 1, 1, 0 ] ], [ 1, NF(24,[ 1, 7 ]), 8, [ 2, 7, 0 ] ],[0X
    [4X  [ 2, CF(3), 12, [ 2, 7, 0 ] ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  In   some   cases   some   of   the   first   entries   of   the  output  of
  [2XWedderburnDecompositionInfo[0m   are  not  integers  and  so  the  correspoding
  [13XWedderburn  components[0m  ([14X7.3[0m)  are  given as "fractional matrix algebras" of
  [13Xcyclotomic  algebras[0m  ([14X7.11[0m).  See [14X7.3[0m for a theoretical explanation of this
  phenomenon.  In  that  case  a  warning message will be displayed during the
  first call of [10XWedderburnDecompositionInfo[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QG:=GroupRing(Rationals,SmallGroup(240,89));[0X
    [4X<algebra-with-one over Rationals, with 2 generators>[0X
    [4Xgap> WedderburnDecompositionInfo(QG);[0X
    [4XWedderga: Warning!!! [0X
    [4XSome of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!![0X
    [4X[0X
    [4X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ],[0X
    [4X  [ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ],[0X
    [4X  [ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ],[0X
    [4X  [ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  The  interpretation  of the output in the previous example gives rise to the
  following  [13XWedderburn  decomposition[0m  ([14X7.3[0m)  of  ℚ  G  for G the small group
  [240,89]:
  
  
       ℚ G = 2 ℚ \oplus 2 M_4( ℚ ) \oplus 2 M_5( ℚ ) \oplus M_6( ℚ )
       \oplus A \oplus B \oplus C
  
  
  where
  
  
       A = ℚ (\xi_{10})[u|\xi_{10}^u = \xi_{10}^3, u^4 = -1],
  
  
  B is an algebra of degree (4*2 )/2 = 4 which is [13XBrauer equivalent[0m ([14X7.5[0m) to
  
  
       B_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^{13}, u^4 =
       \xi_{60}^5, \xi_{60}^v = \xi_{60}^{11}, v^2 = 1, vu=uv],
  
  
  and C is an algebra of degree (4*2)*3/4 = 6 which is [13XBrauer equivalent[0m ([14X7.5[0m)
  to
  
  
       C_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^7, u^4 = \xi_{60}^5,
       \xi_{60}^v = \xi_{60}^{31}, v^2 = 1, vu=uv].
  
  
  The precise description of B and C requires the usage of "ad hoc" arguments.
  
  
  [1X2.2 Simple quotients[0X
  
  [1X2.2-1 SimpleAlgebraByCharacter[0m
  
  [2X> SimpleAlgebraByCharacter( [0X[3XFG, chi[0X[2X ) _____________________________[0Xoperation
  [6XReturns:[0X  A simple algebra.
  
  The  first input [3XFG[0m should be a [13Xsemisimple group algebra[0m ([14X7.2[0m) over a finite
  group G and the second input should be an irreducible character of G.
  
  The  output  is  a  matrix  algebra of a [13Xcyclotomic algebras[0m ([14X7.11[0m) which is
  isomorphic  to  the  unique  [13XWedderburn  component[0m  ([14X7.3[0m)  A of [3XFG[0m such that
  chi(A)ne 0.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> A5 := AlternatingGroup(5);[0X
    [4XAlt( [ 1 .. 5 ] )[0X
    [4Xgap> SimpleAlgebraByCharacter( GroupRing( Rationals , A5 ) , Irr( A5 ) [3] );[0X
    [4X( NF(5,[ 1, 4 ])^[ 3, 3 ] )[0X
    [4Xgap> SimpleAlgebraByCharacter( GroupRing( GF(7) , A5 ) , Irr( A5 ) [3] );[0X
    [4X( GF(7^2)^[ 3, 3 ] )[0X
    [4Xgap> G:=SmallGroup(128,100);[0X
    [4X<pc group of size 128 with 7 generators>[0X
    [4Xgap> SimpleAlgebraByCharacter( GroupRing( Rationals , G ) , Irr(G)[19] );[0X
    [4X<crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[ 1, 3 ]), CF([0X
    [4X8) ) of a group of size 2>[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.2-2 SimpleAlgebraByCharacterInfo[0m
  
  [2X> SimpleAlgebraByCharacterInfo( [0X[3XFG, chi[0X[2X ) _________________________[0Xoperation
  [6XReturns:[0X  The     numerical     description     of     the     output     of
            [2XSimpleAlgebraByCharacter[0m ([14X2.2-1[0m).
  
  The first input [3XFG[0m is a [13Xsemisimple group algebra[0m ([14X7.2[0m) over a finite group G
  and the second input is an irreducible character of G.
  
  The  output  is  the  numerical  description  [14X7.12[0m of the [13Xcyclotomic algebra[0m
  ([14X7.11[0m)  which is isomorphic to the unique [13XWedderburn component[0m ([14X7.3[0m) A of [3XFG[0m
  such that chi(A)ne 0.
  
  See  [14X7.12[0m  for  the interpretation of the numerical information given by the
  output.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> G:=SmallGroup(144,11);[0X
    [4X<pc group of size 144 with 6 generators>[0X
    [4Xgap> QG:=GroupRing(Rationals,G);[0X
    [4X<algebra-with-one over Rationals, with 6 generators>[0X
    [4Xgap> SimpleAlgebraByCharacter( QG , Irr(G)[48] );[0X
    [4X<crossed product with center NF(36,[ 1, 17 ]) over AsField( NF(36,[0X
    [4X[ 1, 17 ]), CF(36) ) of a group of size 2>[0X
    [4Xgap> SimpleAlgebraByCharacterInfo( QG , Irr(G)[48] );[0X
    [4X[ 1, NF(36,[ 1, 17 ]), 36, [ 2, 17, 18 ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.2-3 SimpleAlgebraByStrongSP[0m
  
  [2X> SimpleAlgebraByStrongSP( [0X[3XQG, K, H[0X[2X ) _____________________________[0Xoperation
  [2X> SimpleAlgebraByStrongSPNC( [0X[3XQG, K, H[0X[2X ) ___________________________[0Xoperation
  [2X> SimpleAlgebraByStrongSP( [0X[3XFG, K, H, C[0X[2X ) __________________________[0Xoperation
  [2X> SimpleAlgebraByStrongSPNC( [0X[3XFG, K, H, C[0X[2X ) ________________________[0Xoperation
  [6XReturns:[0X  A simple algebra.
  
  In  the  three-argument  version  the  input  must be formed by a [13Xsemisimple
  rational  group  algebra[0m  [3XQG[0m  (see [14X7.2[0m) and two subgroups [3XK[0m and [3XH[0m of G which
  form a [13Xstrong Shoda pair[0m ([14X7.15[0m) of G.
  
  The  three-argument  version  returns  the Wedderburn component ([14X7.3[0m) of the
  rational group algebra [3XQG[0m realized by the strong Shoda pair ([3XK[0m,[3XH[0m).
  
  In the four-argument version the first argument is a semisimple finite group
  algebra  [3XFG[0m,  [3X(K,H)[0m is a strong Shoda pair of G and the fourth input data is
  either  a  generating  q-cyclotomic  class  modulo  the index of [3XH[0m in [3XK[0m or a
  representative of a generating q-cyclotomic class modulo the index of [3XH[0m in [3XK[0m
  (see [14X7.17[0m).
  
  The  four-argument  version  returns  the  Wedderburn component ([14X7.3[0m) of the
  finite  group  algebra  [3XFG[0m  realized  by the strong Shoda pair ([3XK[0m,[3XH[0m) and the
  cyclotomic class [3XC[0m (or the cyclotomic class containing [3XC[0m).
  
  The versions ending in NC do not check if ([3XK[0m,[3XH[0m) is a strong Shoda pair of G.
  In  the  four-argument  version it is also not checked whether [3XC[0m is either a
  generating  q-cyclotomic  class  modulo  the  index  of [3XH[0m in [3XK[0m or an integer
  coprime to the index of [3XH[0m in [3XK[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;[0X
    [4Xgap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;[0X
    [4Xgap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;[0X
    [4Xgap> QG:=GroupRing( Rationals, G );;[0X
    [4Xgap> FG:=GroupRing( GF(7), G );;[0X
    [4Xgap> SimpleAlgebraByStrongSP( QG, K, H );[0X
    [4X<crossed product over CF(16) of a group of size 2>[0X
    [4Xgap> SimpleAlgebraByStrongSP( FG, K, H, [1,7] );[0X
    [4X( GF(7)^[ 2, 2 ] )[0X
    [4Xgap> SimpleAlgebraByStrongSP( FG, K, H, 1 );[0X
    [4X( GF(7)^[ 2, 2 ] )[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.2-4 SimpleAlgebraByStrongSPInfo[0m
  
  [2X> SimpleAlgebraByStrongSPInfo( [0X[3XQG, K, H[0X[2X ) _________________________[0Xoperation
  [2X> SimpleAlgebraByStrongSPInfoNC( [0X[3XQG, K, H[0X[2X ) _______________________[0Xoperation
  [2X> SimpleAlgebraByStrongSPInfo( [0X[3XFG, K, H, C[0X[2X ) ______________________[0Xoperation
  [2X> SimpleAlgebraByStrongSPInfoNC( [0X[3XFG, K, H, C[0X[2X ) ____________________[0Xoperation
  [6XReturns:[0X  A numerical description of one simple algebra.
  
  In  the  three-argument  version  the  input  must be formed by a [13Xsemisimple
  rational  group algebra[0m ([14X7.2[0m) [3XQG[0m and two subgroups [3XK[0m and [3XH[0m of G which form a
  [13Xstrong  Shoda  pair[0m  ([14X7.15[0m)  of  G.  It  returns  the  numerical information
  describing  the Wedderburn component ([14X7.12[0m) of the rational group algebra [3XQG[0m
  realized by a the strong Shoda pair ([3XK[0m,[3XH[0m).
  
  In  the  four-argument  version the first input is a semisimple finite group
  algebra  [3XFG[0m,  [3X(K,H)[0m is a strong Shoda pair of G and the fourth input data is
  either  a  generating  q-cyclotomic  class  modulo  the index of [3XH[0m in [3XK[0m or a
  representative of a generating q-cyclotomic class modulo the index of [3XH[0m in [3XK[0m
  ([14X7.17[0m).  It returns a pair of positive integers [n,r] which represent the nx
  n  matrix  algebra  over  the  field  of  order r which is isomorphic to the
  Wedderburn component of [3XFG[0m realized by a the strong Shoda pair ([3XK[0m,[3XH[0m) and the
  cyclotomic class [3XC[0m (or the cyclotomic class containing the integer [3XC[0m).
  
  The versions ending in NC do not check if ([3XK[0m,[3XH[0m) is a strong Shoda pair of G.
  In  the  four-argument  version it is also not checked whether [3XC[0m is either a
  generating  q-cyclotomic  class  modulo  the  index  of [3XH[0m in [3XK[0m or an integer
  coprime with the index of [3XH[0m in [3XK[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;[0X
    [4Xgap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;[0X
    [4Xgap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);; [0X
    [4Xgap> QG:=GroupRing( Rationals, G );;[0X
    [4Xgap> FG:=GroupRing( GF(7), G );;[0X
    [4Xgap> SimpleAlgebraByStrongSP( QG, K, H );[0X
    [4X<crossed product over CF(16) of a group of size 2>[0X
    [4Xgap> SimpleAlgebraByStrongSPInfo( QG, K, H );[0X
    [4X[ 1, NF(16,[ 1, 7 ]), 16, [ [ 2, 7, 8 ] ], [  ] ][0X
    [4Xgap> SimpleAlgebraByStrongSPInfo( FG, K, H, [1,7] );[0X
    [4X[ 2, 7 ][0X
    [4Xgap> SimpleAlgebraByStrongSPInfo( FG, K, H, 1 );[0X
    [4X[ 2, 7 ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
