  
  [1m[4m[31m2. Wedderburn decomposition[0m
  
  
  [1m[4m[31m2.1 Wedderburn decomposition[0m
  
  [1m[4m[31m2.1-1 WedderburnDecomposition[0m
  
  [1m[34m> WedderburnDecomposition( [0m[22m[34mFG[0m[1m[34m ) ___________________________________[0mattribute
  [1mReturns:[0m  A list of simple algebras.
  
  The input [22m[34mFG[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 [22m[36mWedderburn components[0m ([1m7.3[0m) of the
  group  algebra  [22m[34mFG[0m.  If  F  is  an abelian number field then each Wedderburn
  component is given as a matrix algebras of a [22m[36mcyclotomic algebra[0m ([1m7.11[0m). If F
  is  a  finite  field  then  the  Wedderburn  components  are given as matrix
  algebras over finite fields.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> WedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) );[0m
    [22m[35m[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ),[0m
    [22m[35m  ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ][0m
    [22m[35mgap> WedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) );[0m
    [22m[35m[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[0m
    [22m[35m  <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[0m
    [22m[35m    [ 1, 7 ]), CF(8) ) of a group of size 2> ][0m
    [22m[35mgap> WedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) );[0m
    [22m[35m[ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ),[0m
    [22m[35m  <crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40,[0m
    [22m[35m    [ 1, 31 ]), CF(40) ) of a group of size 2> ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  The  previous examples show that if D_16 denotes the dihedral group of order
  16  then the [22m[36mWedderburn decomposition[0m ([1m7.3[0m) of F_5 D_16, Q D_16 and Q (xi_5)
  D_16 are respectively
  
  \[
       \F_5 D_{16} = 4 \F_5 \oplus M_2( \F_5 ) \oplus M_2( \F_{25} ),
  \]
  
  \[
       \Q D_{16} = 4 \Q \oplus M_2( \Q ) \oplus (K(\xi_8)/K,t),
  \]
  
  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 [22m[36mcyclotomic algebra[0m ([1m7.11[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:
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );[0m
    [22m[35m[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[0m
    [22m[35m  <crossed product with center Rationals over CF(3) of a group of size 2>,[0m
    [22m[35m  ( CF(3)^[ 2, 2 ] ), <crossed product with center Rationals over CF([0m
    [22m[35m    3) of a group of size 2>, <crossed product with center NF(8,[0m
    [22m[35m    [ 1, 7 ]) over AsField( NF(8,[ 1, 7 ]), CF(8) ) of a group of size 2>,[0m
    [22m[35m  <crossed product with center Rationals over CF(12) of a group of size 4> ][0m
    [22m[35mgap> WedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) );[0m
    [22m[35m[ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ),[0m
    [22m[35m  ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ),[0m
    [22m[35m  <crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24,[0m
    [22m[35m    [ 1, 7 ]), CF(24) ) of a group of size 2>,[0m
    [22m[35m  ( <crossed product with center CF(3) over AsField( CF(3), CF([0m
    [22m[35m    12) ) of a group of size 2>^[ 2, 2 ] ) ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  In  some  cases,  in  characteristic  zero,  some  entries  of the output of
  [1m[34mWedderburnDecomposition[0m   do   not  provide  full  matrix  algebras  over  a
  [22m[36mcyclotomic  algebra[0m  ([1m7.11[0m), but "fractional matrix algebras". That entry is
  not an algebra that can be used as a [1mGAP[0m object. Instead is a pair form by a
  rational  giving  the  "size" of the matrices and a crossed product. See [1m7.3[0m
  for  a  theoretical  explanation  of this phenomenon. In this case a warning
  message is displayed.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> QG:=GroupRing(Rationals,SmallGroup(240,89));[0m
    [22m[35m<algebra-with-one over Rationals, with 2 generators>[0m
    [22m[35mgap> WedderburnDecomposition(QG);[0m
    [22m[35mWedderga: Warning!!![0m
    [22m[35mSome of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!![0m
    [22m[35m[0m
    [22m[35m[ Rationals, Rationals, <crossed product with center Rationals over CF([0m
    [22m[35m    5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ),[0m
    [22m[35m  ( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ),[0m
    [22m[35m  <crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12,[0m
    [22m[35m    [ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>,[0m
    [22m[35m  [ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[0m
    [22m[35m        [ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ]  [0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  [1m[4m[31m2.1-2 WedderburnDecompositionInfo[0m
  
  [1m[34m> WedderburnDecompositionInfo( [0m[22m[34mFG[0m[1m[34m ) _______________________________[0mattribute
  [1mReturns:[0m  A  list  with  each  entry a numerical description of a [22m[36mcyclotomic
            algebras[0m ([1m7.11[0m).
  
  The input [22m[34mFG[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.
  
  This function is a numerical counterpart of [1m[34mWedderburnDecomposition[0m ([1m2.1-1[0m).
  
  It returns a list formed by lists of length 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 ntimes 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  conditions mentioned in section [1m7.12[0m. It represents the ntimes 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  ntimes  n  matrix  algebra  M_n(A)  over the
  [22m[36mcyclotomic algebra[0m ([1m7.11[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 [1m7.12[0m).
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> WedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) );[0m
    [22m[35m[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],[0m
    [22m[35m  [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ] ][0m
    [22m[35mgap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) );[0m
    [22m[35m[ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ],[0m
    [22m[35m  [ 1, NF(40,[ 1, 31 ]), 8, [ 2, 7, 0 ] ] ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  The  interpretation  of  the  previous  example  gives rise to the following
  [22m[36mWedderburn  decompositions[0m  ([1m7.3[0m), where D_16 is the dihedral group of order
  16 and xi_5 is a primitive 5-th root of unity.
  
  \[
       \Q D_{16} = 4 \Q \oplus M_2( \Q ) \oplus M_2( \Q (\sqrt{2})).
  \]
  
  \[
       \Q (\xi_5) D_{16} = 4 \Q (\xi_5) \oplus M_2( \Q (\xi_5)) \oplus
       M_2( \Q (\xi_5,\sqrt{2})).
  \]
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;;rel:=[a^8,a^4*b^2,b^-1*a*b*a];;[0m
    [22m[35mgap> Q16:=F/rel;; QQ16:=GroupRing( Rationals, Q16 );;[0m
    [22m[35mgap> QS4:=GroupRing( Rationals, SymmetricGroup(4) );;[0m
    [22m[35mgap> WedderburnDecomposition(QQ16);[0m
    [22m[35m[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[0m
    [22m[35m  <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[0m
    [22m[35m    [ 1, 7 ]), CF(8) ) of a group of size 2> ][0m
    [22m[35mgap> WedderburnDecomposition( QS4 );[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> WedderburnDecompositionInfo(QQ16);[0m
    [22m[35m[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [0m
    [22m[35m  [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ] ][0m
    [22m[35mgap> WedderburnDecompositionInfo(QS4);  [0m
    [22m[35m[ [ 1, Rationals ], [ 1, Rationals ], [ 3, Rationals ], [ 3, Rationals ], [0m
    [22m[35m  [ 1, Rationals, 3, [ 2, 2, 0 ] ] ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  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  [1m[34mWedderburnDecomposition[0m  ([1m2.1-1[0m)  and
  [1m[34mWedderburnDecompositionInfo[0m.
  
  The output of [1m[34mWedderburnDecomposition[0m ([1m2.1-1[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 [22m[36mcrossed products[0m ([1m7.6[0m) with coefficients in the cyclotomic
  fields  Q (xi_8) and Q (xi_3) respectively. This output can be used as a [1mGAP[0m
  object,  but it does not give clear information on the structure of algebras
  A and B.
  
  The  numerical  information  displayed  by [1m[34mWedderburnDecompositionInfo[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. [R03]).
  
  Next  example  shows the output of [22m[32mWedderburnDecompositionInfo[0m for Q G and Q
  (xi_3) G, where G=SmallGroup(48,15). The user can compare it with the output
  of  [1m[34mWedderburnDecomposition[0m  ([1m2.1-1[0m)  for  the  same  group  in the previous
  section. Notice that the last entry of the [22m[36mWedderburn decomposition[0m ([1m7.3[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.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> WedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) );[0m
    [22m[35m[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],[0m
    [22m[35m  [ 2, Rationals ], [ 1, Rationals, 3, [ 2, 2, 0 ] ], [ 2, CF(3) ],[0m
    [22m[35m  [ 1, Rationals, 6, [ 2, 5, 0 ] ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ],[0m
    [22m[35m  [ 1, Rationals, 12, [ [ 2, 5, 9 ], [ 2, 7, 0 ] ], [ [ 9 ] ] ] ][0m
    [22m[35mgap> WedderburnDecompositionInfo( GroupRing( CF(3), SmallGroup(48,15) ) );[0m
    [22m[35m[ [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 2, CF(3) ],[0m
    [22m[35m  [ 2, CF(3), 3, [ 1, 1, 0 ] ], [ 2, CF(3) ], [ 2, CF(3) ],[0m
    [22m[35m  [ 2, CF(3), 6, [ 1, 1, 0 ] ], [ 1, NF(24,[ 1, 7 ]), 8, [ 2, 7, 0 ] ],[0m
    [22m[35m  [ 2, CF(3), 12, [ 2, 7, 0 ] ] ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  In   some   cases   some   of   the   first   entries   of   the  output  of
  [1m[34mWedderburnDecompositionInfo[0m   are  not  integers  and  so  the  correspoding
  [22m[36mWedderburn  components[0m  ([1m7.3[0m)  are  given as "fractional matrix algebras" of
  [22m[36mcyclotomic  algebras[0m  ([1m7.11[0m).  See [1m7.3[0m for a theoretical explanation of this
  phenomenon.  In  that  case  a  warning message will be displayed during the
  first call of [22m[32mWedderburnDecompositionInfo[0m.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> QG:=GroupRing(Rationals,SmallGroup(240,89));[0m
    [22m[35m<algebra-with-one over Rationals, with 2 generators>[0m
    [22m[35mgap> WedderburnDecompositionInfo(QG);[0m
    [22m[35mWedderga: Warning!!! [0m
    [22m[35mSome of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!![0m
    [22m[35m[0m
    [22m[35m[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ],[0m
    [22m[35m  [ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ],[0m
    [22m[35m  [ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ],[0m
    [22m[35m  [ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  The  interpretation  of  the output in the previous example give rise to the
  following  [22m[36mWedderburn  decomposition[0m  ([1m7.3[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 [22m[36mBrauer equivalent[0m ([1m7.5[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 [22m[36mBrauer equivalent[0m ([1m7.5[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.
  
  
  [1m[4m[31m2.2 Simple quotients[0m
  
  [1m[4m[31m2.2-1 SimpleAlgebraByCharacter[0m
  
  [1m[34m> SimpleAlgebraByCharacter( [0m[22m[34mFG, chi[0m[1m[34m ) _____________________________[0moperation
  [1mReturns:[0m  A simple algebra.
  
  The  first input [22m[34mFG[0m should be a [22m[36msemisimple group algebra[0m ([1m7.2[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 [22m[36mcyclotomic algebras[0m ([1m7.11[0m) which is
  isomorphic  to  the  unique  [22m[36mWedderburn  component[0m  ([1m7.3[0m)  A of [22m[34mFG[0m such that
  chi(A)ne 0.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> A5 := AlternatingGroup(5);[0m
    [22m[35mAlt( [ 1 .. 5 ] )[0m
    [22m[35mgap> SimpleAlgebraByCharacter( GroupRing( Rationals , A5 ) , Irr( A5 ) [3] );[0m
    [22m[35m( NF(5,[ 1, 4 ])^[ 3, 3 ] )[0m
    [22m[35mgap> SimpleAlgebraByCharacter( GroupRing( GF(7) , A5 ) , Irr( A5 ) [3] );[0m
    [22m[35m( GF(7^2)^[ 3, 3 ] )[0m
    [22m[35mgap> G:=SmallGroup(128,100);[0m
    [22m[35m<pc group of size 128 with 7 generators>[0m
    [22m[35mgap> SimpleAlgebraByCharacter( GroupRing( Rationals , G ) , Irr(G)[19] );[0m
    [22m[35m<crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[ 1, 3 ]), CF([0m
    [22m[35m8) ) of a group of size 2>[0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  [1m[4m[31m2.2-2 SimpleAlgebraByCharacterInfo[0m
  
  [1m[34m> SimpleAlgebraByCharacterInfo( [0m[22m[34mFG, chi[0m[1m[34m ) _________________________[0moperation
  [1mReturns:[0m  The     numerical     description     of     the     output     of
            [1m[34mSimpleAlgebraByCharacter[0m ([1m2.2-1[0m).
  
  The first input [22m[34mFG[0m is a [22m[36msemisimple group algebra[0m ([1m7.2[0m) over a finite group G
  and the second input is an irreducible character of G.
  
  The  output  is  the  numerical  description  [1m7.12[0m of the [22m[36mcyclotomic algebra[0m
  ([1m7.11[0m)  which is isomorphic to the unique [22m[36mWedderburn component[0m ([1m7.3[0m) A of [22m[34mFG[0m
  such that chi(A)ne 0.
  
  See  [1m7.12[0m  for  the interpretation of the numerical information given by the
  output.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> G:=SmallGroup(144,11);[0m
    [22m[35m<pc group of size 144 with 6 generators>[0m
    [22m[35mgap> QG:=GroupRing(Rationals,G);[0m
    [22m[35m<algebra-with-one over Rationals, with 6 generators>[0m
    [22m[35mgap> SimpleAlgebraByCharacter( QG , Irr(G)[48] );[0m
    [22m[35m<crossed product with center NF(36,[ 1, 17 ]) over AsField( NF(36,[0m
    [22m[35m[ 1, 17 ]), CF(36) ) of a group of size 2>[0m
    [22m[35mgap> SimpleAlgebraByCharacterInfo( QG , Irr(G)[48] );[0m
    [22m[35m[ 1, NF(36,[ 1, 17 ]), 36, [ 2, 17, 18 ] ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  [1m[4m[31m2.2-3 SimpleAlgebraByStrongSP[0m
  
  [1m[34m> SimpleAlgebraByStrongSP( [0m[22m[34mQG, K, H[0m[1m[34m ) _____________________________[0moperation
  [1m[34m> SimpleAlgebraByStrongSPNC( [0m[22m[34mQG, K, H[0m[1m[34m ) ___________________________[0moperation
  [1m[34m> SimpleAlgebraByStrongSP( [0m[22m[34mFG, K, H, C[0m[1m[34m ) __________________________[0moperation
  [1m[34m> SimpleAlgebraByStrongSPNC( [0m[22m[34mFG, K, H, C[0m[1m[34m ) ________________________[0moperation
  [1mReturns:[0m  A simple algebra.
  
  In  the  three-argument  version  the  input  must be formed by a [22m[36msemisimple
  rational  group  algebra[0m  [22m[34mQG[0m  (see [1m7.2[0m) and two subgroups [22m[34mK[0m and [22m[34mH[0m of G which
  form a [22m[36mstrong Shoda pair[0m ([1m7.15[0m) of G.
  
  The  three-argument  version  returns  the Wedderburn component ([1m7.3[0m) of the
  rational group algebra [22m[34mQG[0m realized by the strong Shoda pair ([22m[34mK[0m,[22m[34mH[0m).
  
  In the four-argument version the first argument is a semisimple finite group
  algebra  [22m[34mFG[0m,  [22m[34m(K,H)[0m is a strong Shoda pair of G and the fourth input data is
  either  a  generating q-cyclotomic class modulo [[22m[34mK[0m,[22m[34mH[0m] or a representative of
  generating q-cyclotomic class modulo [[22m[34mK[0m,[22m[34mH[0m] (see [1m7.17[0m).
  
  The  four-argument  version  returns  the  Wedderburn component ([1m7.3[0m) of the
  finite  group  algebra  [22m[34mFG[0m  realized  by the strong Shoda pair ([22m[34mK[0m,[22m[34mH[0m) and the
  cyclotomic class [22m[34mC[0m (or the cyclotomic class containing [22m[34mC[0m).
  
  The  versions  ended in NC does not check if ([22m[34mK[0m,[22m[34mH[0m) is a strong Shoda pair of
  G. In the four-argument version it is also not checked whether [22m[34mC[0m is either a
  generating q-cyclotomic class the index of [22m[34mH[0m in [22m[34mK[0m or an integer coprime with
  the index of [22m[34mH[0m in [22m[34mK[0m.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;[0m
    [22m[35mgap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;[0m
    [22m[35mgap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;[0m
    [22m[35mgap> QG:=GroupRing( Rationals, G );;[0m
    [22m[35mgap> FG:=GroupRing( GF(7), G );;[0m
    [22m[35mgap> SimpleAlgebraByStrongSP( QG, K, H );[0m
    [22m[35m<crossed product over CF(16) of a group of size 2>[0m
    [22m[35mgap> SimpleAlgebraByStrongSP( FG, K, H, [1,7] );[0m
    [22m[35m( GF(7)^[ 2, 2 ] )[0m
    [22m[35mgap> SimpleAlgebraByStrongSP( FG, K, H, 1 );[0m
    [22m[35m( GF(7)^[ 2, 2 ] )[0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  [1m[4m[31m2.2-4 SimpleAlgebraByStrongSPInfo[0m
  
  [1m[34m> SimpleAlgebraByStrongSPInfo( [0m[22m[34mQG, K, H[0m[1m[34m ) _________________________[0moperation
  [1m[34m> SimpleAlgebraByStrongSPInfoNC( [0m[22m[34mQG, K, H[0m[1m[34m ) _______________________[0moperation
  [1m[34m> SimpleAlgebraByStrongSPInfo( [0m[22m[34mFG, K, H, C[0m[1m[34m ) ______________________[0moperation
  [1m[34m> SimpleAlgebraByStrongSPInfoNC( [0m[22m[34mFG, K, H, C[0m[1m[34m ) ____________________[0moperation
  [1mReturns:[0m  A numerical description of one simple algebra.
  
  In  the  three-argument  version  the  input  must be formed by a [22m[36msemisimple
  rational  group algebra[0m ([1m7.2[0m) [22m[34mQG[0m and two subgroups [22m[34mK[0m and [22m[34mH[0m of G which form a
  [22m[36mstrong  Shoda  pair[0m  ([1m7.15[0m)  of  G.  It  returns  the  numerical information
  describing  the Wedderburn component ([1m7.12[0m) of the rational group algebra [22m[34mQG[0m
  realized by a the strong Shoda pair ([22m[34mK[0m,[22m[34mH[0m).
  
  In  the  four-argument  version the first input is a semisimple finite group
  algebra  [22m[34mFG[0m,  [22m[34m(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 [22m[34mH[0m in [22m[34mK[0m or a
  representative  of  generating q-cyclotomic class modulo the index of [22m[34mH[0m in [22m[34mK[0m
  ([1m7.17[0m).  It  returns  a  pair of positive integers [n,r] which represent the
  ntimes n matrix algebra over the field of order r which is isomorphic to the
  Wedderburn component of [22m[34mFG[0m realized by a the strong Shoda pair ([22m[34mK[0m,[22m[34mH[0m) and the
  cyclotomic class [22m[34mC[0m (or the cyclotomic class containing the integer [22m[34mC[0m).
  
  The  versions  ended in NC does not check if ([22m[34mK[0m,[22m[34mH[0m) is a strong Shoda pair of
  G. In the four-argument version it is also not checked whether [22m[34mC[0m is either a
  generating  q-cyclotomic  class  modulo  the  index  of [22m[34mH[0m in [22m[34mK[0m or an integer
  coprime with the index of [22m[34mH[0m in [22m[34mK[0m.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;[0m
    [22m[35mgap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;[0m
    [22m[35mgap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);; [0m
    [22m[35mgap> QG:=GroupRing( Rationals, G );;[0m
    [22m[35mgap> FG:=GroupRing( GF(7), G );;[0m
    [22m[35mgap> SimpleAlgebraByStrongSP( QG, K, H );[0m
    [22m[35m<crossed product over CF(16) of a group of size 2>[0m
    [22m[35mgap> SimpleAlgebraByStrongSPInfo( QG, K, H );[0m
    [22m[35m[ 1, NF(16,[ 1, 7 ]), 16, [ [ 2, 7, 8 ] ], [  ] ][0m
    [22m[35mgap> SimpleAlgebraByStrongSPInfo( FG, K, H, [1,7] );[0m
    [22m[35m[ 2, 7 ][0m
    [22m[35mgap> SimpleAlgebraByStrongSPInfo( FG, K, H, 1 );[0m
    [22m[35m[ 2, 7 ][0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
