[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, â D_16 and â (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} ), â D_{16} = 4 â \oplus M_2( â ) \oplus (K(\xi_8)/K,t), and â (\xi_5) D_{16} = 4 â (\xi_5) \oplus M_2( â (\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 ])= â (sqrt2), (F(xi_40)/F,t) = â (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. â D_{16} = 4 â \oplus M_2( â ) \oplus M_2( â (\sqrt{2})). â (\xi_5) D_{16} = 4 â (\xi_5) \oplus M_2( â (\xi_5)) \oplus M_2( â (\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 â Q_16 of the quaternion group of order 16 and the rational group algebra â 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 â Q_{16} = 4 â \oplus M_2( â ) \oplus A, â S_{4} = 2 â \oplus 2 M_3( â ) \oplus B, where A and B are [13Xcrossed products[0m ([14X7.6[0m) with coefficients in the cyclotomic fields â (xi_8) and â (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 = â (\xi|\xi^8=1)[g | \xi^g = \xi^7 = \xi^{-1}, g^2 = \xi^4 = -1], B = â (\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 â (xi+xi^-1) and the former is equal to â (sqrt2) and â respectively. In B, one has (g+1)(g-1)=0, while g is neither 1 nor -1. This shows that B=M_2( â ). However the relation g^2=-1 in A shows that A=â (\sqrt{2})[i,g|i^2=g^2=-1,ig=-gi] and so A is a division algebra with centre â (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 â G and â (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 â G is not given as a matrix algebra of a cyclic algebra. However, the corresponding entry of â (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 â G for G the small group [240,89]: â G = 2 â \oplus 2 M_4( â ) \oplus 2 M_5( â ) \oplus M_6( â ) \oplus A \oplus B \oplus C where A = â (\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 = â (\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 = â (\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