<!-- $Id: decomp.xml,v 1.23 2007/12/27 19:33:26 alexk Exp $ --> <!-- ********************Decomp******************** --> <Chapter Label="Decomp"> <Heading>Wedderburn decomposition</Heading> <Section Label="DecompDecomp"> <Heading>Wedderburn decomposition</Heading> <ManSection> <Attr Name="WedderburnDecomposition" Arg="FG" Comm="The Wedderburn decomposition of a group algebras given as a list of GAP objects" /> <Returns> A list of simple algebras. </Returns> <Description> The input <A>FG</A> should be a group algebra of a finite group <M>G</M> over the field <M>F</M>, where <M>F</M> 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 <M>G</M>. <P/> The function returns the list of all <E>Wedderburn components</E> (<Ref Sect="WedDec" />) of the group algebra <A>FG</A>. If <M>F</M> is an abelian number field then each Wedderburn component is given as a matrix algebra of a <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />). If <M>F</M> is a finite field then the Wedderburn components are given as matrix algebras over finite fields. <Example> <![CDATA[ gap> WedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) ); [ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ] gap> WedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) ); [ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ), <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8, [ 1, 7 ]), CF(8) ) of a group of size 2> ] gap> WedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) ); [ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ), <crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40, [ 1, 31 ]), CF(40) ) of a group of size 2> ] ]]> </Example> </Description> </ManSection> The previous examples show that if <M>D_{16}</M> denotes the dihedral group of order <M>16</M> then the <E>Wedderburn decomposition</E> (<Ref Sect="WedDec"/>) of <M>\mathbb F_5 D_{16}</M>, <M>&QQ; D_{16}</M> and <M>&QQ; (\xi_5) D_{16}</M> are respectively <Display> \mathbb F_5 D_{16} = 4 \mathbb F_5 \oplus M_2( \mathbb F_5 ) \oplus M_2( \mathbb F_{25} ), </Display> <Display> &QQ; D_{16} = 4 &QQ; \oplus M_2( &QQ; ) \oplus (K(\xi_8)/K,t), </Display> and <Display> &QQ; (\xi_5) D_{16} = 4 &QQ; (\xi_5) \oplus M_2( &QQ; (\xi_5) ) \oplus (F(\xi_{40})/F,t), </Display> where <M>(K(\xi_8)/K,t)</M> is a <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />) with the centre <M>K=NF(8,[ 1, 7 ])= &QQ; (\sqrt{2})</M>, <M>(F(\xi_{40})/F,t) = &QQ; (\sqrt{2},\xi_5)</M> is a cyclotomic algebra with centre <M>F=NF(40,[ 1, 31 ])</M> and <M>\xi_n</M> denotes a <M>n</M>-th root of unity. <P/> Two more examples: <Example> <![CDATA[ gap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) ); [ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ), <crossed product with center Rationals over CF(3) of a group of size 2>, ( CF(3)^[ 2, 2 ] ), <crossed product with center Rationals over CF( 3) of a group of size 2>, <crossed product with center NF(8, [ 1, 7 ]) over AsField( NF(8,[ 1, 7 ]), CF(8) ) of a group of size 2>, <crossed product with center Rationals over CF(12) of a group of size 4> ] gap> WedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) ); [ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), <crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24, [ 1, 7 ]), CF(24) ) of a group of size 2>, ( <crossed product with center CF(3) over AsField( CF(3), CF( 12) ) of a group of size 2>^[ 2, 2 ] ) ] ]]> </Example> In some cases, in characteristic zero, some entries of the output of <Ref Attr="WedderburnDecomposition" /> do not provide full matrix algebras over a <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />), but "fractional matrix algebras". That entry is not an algebra that can be used as a &GAP; object. Instead it is a pair formed by a rational giving the "size" of the matrices and a crossed product. See <Ref Sect="WedDec" /> for a theoretical explanation of this phenomenon. In this case a warning message is displayed. <Example> <![CDATA[ gap> QG:=GroupRing(Rationals,SmallGroup(240,89)); <algebra-with-one over Rationals, with 2 generators> gap> WedderburnDecomposition(QG); Wedderga: Warning!!! Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!! [ Rationals, Rationals, <crossed product with center Rationals over CF( 5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ), <crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12, [ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>, [ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8, [ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ] ]]> </Example> <!-- ************************ --> <ManSection> <Attr Name="WedderburnDecompositionInfo" Arg="FG" Comm="The Wedderburn decomposition of a group algebras given as a list of lists of numerical data describing each Wedderburn component" /> <Returns> A list with each entry a numerical description of a <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />). </Returns> <Description> The input <A>FG</A> should be a group algebra of a finite group <M>G</M> over the field <M>F</M>, where <M>F</M> 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 <M>G</M>. <P/> This function is a numerical counterpart of <Ref Attr="WedderburnDecomposition" />. <P/> It returns a list formed by lists of lengths 2, 4 or 5. <P/> The lists of length 2 are of the form <Display> [n,F], </Display> where <M>n</M> is a positive integer and <M>F</M> is a field. It represents the <M>n\times n</M> matrix algebra <M>M_n(F)</M> over the field <M>F</M>.<P/> The lists of length 4 are of the form <Display> [n,F,k,[d,\alpha,\beta]], </Display> where <M>F</M> is a field and <M>n,k,d,\alpha,\beta</M> are non-negative integers, satisfying the conditions mentioned in Section <Ref Sect="NumDesc" />. It represents the <M>n\times n</M> matrix algebra <M>M_n(A)</M> over the cyclic algebra <Display> A=F(\xi_k)[u | \xi_k^u = \xi_k^{\alpha}, u^d = \xi_k^{\beta}], </Display> where <M>\xi_k</M> is a primitive <M>k</M>-th root of unity. <P/> The lists of length 5 are of the form <Display> [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ], </Display> where <M>F</M> is a field and <M>n,k,d_i,\alpha_i,\beta_i,\gamma_{i,j}</M> are non-negative integers. It represents the <M>n\times n</M> matrix algebra <M>M_n(A)</M> over the <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />) <Display> 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], </Display> where <M>\xi_k</M> is a primitive <M>k</M>-th root of unity (see <Ref Sect="NumDesc" />). <Example> <![CDATA[ gap> WedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) ); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ] ] gap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) ); [ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ], [ 1, NF(40,[ 1, 31 ]), 8, [ 2, 7, 0 ] ] ] ]]> </Example> </Description> </ManSection> The interpretation of the previous example gives rise to the following <E>Wedderburn decompositions</E> (<Ref Sect="WedDec" />), where <M>D_{16}</M> is the dihedral group of order 16 and <M>\xi_5</M> is a primitive <M>5</M>-th root of unity. <Display> &QQ; D_{16} = 4 &QQ; \oplus M_2( &QQ; ) \oplus M_2( &QQ; (\sqrt{2})). </Display> <Display> &QQ; (\xi_5) D_{16} = 4 &QQ; (\xi_5) \oplus M_2( &QQ; (\xi_5)) \oplus M_2( &QQ; (\xi_5,\sqrt{2})). </Display> <Example> <![CDATA[ gap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;;rel:=[a^8,a^4*b^2,b^-1*a*b*a];; gap> Q16:=F/rel;; QQ16:=GroupRing( Rationals, Q16 );; gap> QS4:=GroupRing( Rationals, SymmetricGroup(4) );; gap> WedderburnDecomposition(QQ16); [ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ), <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8, [ 1, 7 ]), CF(8) ) of a group of size 2> ] gap> WedderburnDecomposition( QS4 ); [ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ), <crossed product with center Rationals over CF(3) of a group of size 2> ] gap> WedderburnDecompositionInfo(QQ16); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ] ] gap> WedderburnDecompositionInfo(QS4); [ [ 1, Rationals ], [ 1, Rationals ], [ 3, Rationals ], [ 3, Rationals ], [ 1, Rationals, 3, [ 2, 2, 0 ] ] ] ]]> </Example> In the previous example we computed the Wedderburn decomposition of the rational group algebra <M>&QQ; Q_{16}</M> of the quaternion group of order <M>16</M> and the rational group algebra <M>&QQ; S_{4}</M> of the symmetric group on four letters. For the two group algebras we used both <Ref Attr="WedderburnDecomposition" /> and <Ref Attr="WedderburnDecompositionInfo" />. <P/> The output of <Ref Attr="WedderburnDecomposition" /> shows that <Display> &QQ; Q_{16} = 4 &QQ; \oplus M_2( &QQ; ) \oplus A, </Display> <Display> &QQ; S_{4} = 2 &QQ; \oplus 2 M_3( &QQ; ) \oplus B, </Display> where <M>A</M> and <M>B</M> are <E>crossed products</E> (<Ref Sect="CrossedProd" />) with coefficients in the cyclotomic fields <M>&QQ; (\xi_8)</M> and <M>&QQ; (\xi_3)</M> respectively. This output can be used as a &GAP; object, but it does not give clear information on the structure of the algebras <M>A</M> and <M>B</M>. <P/> The numerical information displayed by <Ref Attr="WedderburnDecompositionInfo" /> means that <Display> A = &QQ; (\xi|\xi^8=1)[g | \xi^g = \xi^7 = \xi^{-1}, g^2 = \xi^4 = -1], </Display> <Display> B = &QQ; (\xi|\xi^3=1)[g | \xi^g = \xi^2 = \xi^{-1}, g^2 = 1]. </Display> Both <M>A</M> and <M>B</M> are quaternion algebras over its centre which is <M>&QQ; (\xi+\xi^{-1})</M> and the former is equal to <M>&QQ; (\sqrt{2})</M> and <M>&QQ;</M> respectively. <P/> In <M>B</M>, one has <M>(g+1)(g-1)=0</M>, while <M>g</M> is neither <M>1</M> nor <M>-1</M>. This shows that <M>B=M_2( &QQ; )</M>. However the relation <M>g^2=-1</M> in <M>A</M> shows that <Display> A=&QQ; (\sqrt{2})[i,g|i^2=g^2=-1,ig=-gi] </Display> and so <M>A</M> is a division algebra with centre <M>&QQ; (\sqrt{2})</M>, 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. <Cite Key="R" />). <P/> The next example shows the output of <C>WedderburnDecompositionInfo</C> for <M>&QQ; G</M> and <M>&QQ; (\xi_3) G</M>, where <M>G=SmallGroup(48,15)</M>. The user can compare it with the output of <Ref Attr="WedderburnDecomposition" /> for the same group in the previous section. Notice that the last entry of the <E>Wedderburn decomposition</E> (<Ref Sect="WedDec" />) of <M>&QQ; G</M> is not given as a matrix algebra of a cyclic algebra. However, the corresponding entry of <M>&QQ; (\xi_3) G</M> is a matrix algebra of a cyclic algebra. <Example> <![CDATA[ gap> WedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) ); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 1, Rationals, 3, [ 2, 2, 0 ] ], [ 2, CF(3) ], [ 1, Rationals, 6, [ 2, 5, 0 ] ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ], [ 1, Rationals, 12, [ [ 2, 5, 9 ], [ 2, 7, 0 ] ], [ [ 9 ] ] ] ] gap> WedderburnDecompositionInfo( GroupRing( CF(3), SmallGroup(48,15) ) ); [ [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 2, CF(3) ], [ 2, CF(3), 3, [ 1, 1, 0 ] ], [ 2, CF(3) ], [ 2, CF(3) ], [ 2, CF(3), 6, [ 1, 1, 0 ] ], [ 1, NF(24,[ 1, 7 ]), 8, [ 2, 7, 0 ] ], [ 2, CF(3), 12, [ 2, 7, 0 ] ] ] ]]> </Example> In some cases some of the first entries of the output of <Ref Attr="WedderburnDecompositionInfo" /> are not integers and so the correspoding <E>Wedderburn components</E> (<Ref Sect="WedDec" />) are given as "fractional matrix algebras" of <E>cyclotomic algebras</E> (<Ref Sect="Cyclotomic" />). See <Ref Sect="WedDec" /> for a theoretical explanation of this phenomenon. In that case a warning message will be displayed during the first call of <C>WedderburnDecompositionInfo</C>. <Example> <![CDATA[ gap> QG:=GroupRing(Rationals,SmallGroup(240,89)); <algebra-with-one over Rationals, with 2 generators> gap> WedderburnDecompositionInfo(QG); Wedderga: Warning!!! Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!! [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ], [ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ], [ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ], [ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ] ]]> </Example> The interpretation of the output in the previous example gives rise to the following <E>Wedderburn decomposition</E> (<Ref Sect="WedDec" />) of <M>&QQ; G</M> for <M>G</M> the small group <M>[240,89]</M>: <Display> &QQ; G = 2 &QQ; \oplus 2 M_4( &QQ; ) \oplus 2 M_5( &QQ; ) \oplus M_6( &QQ; ) \oplus A \oplus B \oplus C </Display> where <Display> A = &QQ; (\xi_{10})[u|\xi_{10}^u = \xi_{10}^3, u^4 = -1], </Display> <M>B</M> is an algebra of degree <M>(4*2 )/2 = 4 </M> which is <E>Brauer equivalent</E> (<Ref Sect="Brauer" />) to <Display> B_1 = &QQ; (\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], </Display> and <M>C</M> is an algebra of degree <M>(4*2)*3/4 = 6 </M> which is <E>Brauer equivalent</E> (<Ref Sect="Brauer" />) to <Display> C_1 = &QQ; (\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]. </Display> The precise description of <M>B</M> and <M>C</M> requires the usage of "ad hoc" arguments. </Section> <!-- ********************************************************* --> <Section Label="DecompSimple"> <Heading>Simple quotients</Heading> <ManSection> <Oper Name="SimpleAlgebraByCharacter" Arg="FG chi" Comm="The Wedderburn component of the group algebra FG determined by the irreducible character chi" /> <Returns> A simple algebra. </Returns> <Description> The first input <A>FG</A> should be a <E>semisimple group algebra</E> (<Ref Sect="Semisimple" />) over a finite group <M>G</M> and the second input should be an irreducible character of <M>G</M>.<P/> The output is a matrix algebra of a <E>cyclotomic algebras</E> (<Ref Sect="Cyclotomic" />) which is isomorphic to the unique <E>Wedderburn component</E> (<Ref Sect="WedDec" />) <M>A</M> of <A>FG</A> such that <M>\chi(A)\ne 0</M>. <Example> <![CDATA[ gap> A5 := AlternatingGroup(5); Alt( [ 1 .. 5 ] ) gap> SimpleAlgebraByCharacter( GroupRing( Rationals , A5 ) , Irr( A5 ) [3] ); ( NF(5,[ 1, 4 ])^[ 3, 3 ] ) gap> SimpleAlgebraByCharacter( GroupRing( GF(7) , A5 ) , Irr( A5 ) [3] ); ( GF(7^2)^[ 3, 3 ] ) gap> G:=SmallGroup(128,100); <pc group of size 128 with 7 generators> gap> SimpleAlgebraByCharacter( GroupRing( Rationals , G ) , Irr(G)[19] ); <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[ 1, 3 ]), CF( 8) ) of a group of size 2> ]]> </Example> </Description> </ManSection> <!-- ************************ --> <ManSection> <Oper Name="SimpleAlgebraByCharacterInfo" Arg="FG chi" Comm="The numerical data describing the output of SimpleAlgebraByCharacter" /> <Returns> The numerical description of the output of <Ref Attr="SimpleAlgebraByCharacter" />. </Returns> <Description> The first input <A>FG</A> is a <E>semisimple group algebra</E> (<Ref Sect="Semisimple" />) over a finite group <M>G</M> and the second input is an irreducible character of <M>G</M>. <P/> The output is the numerical description <Ref Sect="NumDesc" /> of the <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />) which is isomorphic to the unique <E>Wedderburn component</E> (<Ref Sect="WedDec" />) <M>A</M> of <A>FG</A> such that <M>\chi(A)\ne 0</M>. <P/> See <Ref Sect="NumDesc" /> for the interpretation of the numerical information given by the output. <Example> <![CDATA[ gap> G:=SmallGroup(144,11); <pc group of size 144 with 6 generators> gap> QG:=GroupRing(Rationals,G); <algebra-with-one over Rationals, with 6 generators> gap> SimpleAlgebraByCharacter( QG , Irr(G)[48] ); <crossed product with center NF(36,[ 1, 17 ]) over AsField( NF(36, [ 1, 17 ]), CF(36) ) of a group of size 2> gap> SimpleAlgebraByCharacterInfo( QG , Irr(G)[48] ); [ 1, NF(36,[ 1, 17 ]), 36, [ 2, 17, 18 ] ] ]]> </Example> </Description> </ManSection> <!-- ************************ --> <ManSection> <Oper Name="SimpleAlgebraByStrongSP" Label="for rational group algebra" Arg="QG K H" Comm="The Wedderburn component of the rational group algebra QG realized by (K,H), a strong Shoda pair of G" /> <Oper Name="SimpleAlgebraByStrongSPNC" Label="for rational group algebra" Arg="QG K H" Comm="Same as the non NC version without checking that (K,H) is a strong Shoda pair of G" /> <Oper Name="SimpleAlgebraByStrongSP" Label="for semisimple finite group algebra" Arg="FG K H C" Comm="The Wedderburn component of the semisimple finite group algebra FG given realized by (K,H), a strong Shoda pair of G, and a q-cyclotomic class" /> <Oper Name="SimpleAlgebraByStrongSPNC" Label="for semisimple finite group algebra" Arg="FG K H C" Comm="Same as the non NC version without checking that (K,H) is a strong Shoda pair of G" /> <Returns> A simple algebra. </Returns> <Description> In the three-argument version the input must be formed by a <E>semisimple rational group algebra</E> <A>QG</A> (see <Ref Sect="Semisimple" />) and two subgroups <A>K</A> and <A>H</A> of <M>G</M> which form a <E>strong Shoda pair</E> (<Ref Sect="SSPDef" />) of <M>G</M>. <P/> The three-argument version returns the Wedderburn component (<Ref Sect="WedDec" />) of the rational group algebra <A>QG</A> realized by the strong Shoda pair (<A>K</A>,<A>H</A>). <P/> In the four-argument version the first argument is a semisimple finite group algebra <A>FG</A>, <A>(K,H)</A> is a strong Shoda pair of <M>G</M> and the fourth input data is either a generating <M>q</M>-cyclotomic class modulo the index of <A>H</A> in <A>K</A> or a representative of a generating <M>q</M>-cyclotomic class modulo the index of <A>H</A> in <A>K</A> (see <Ref Sect="CyclotomicClass" />). <P/> The four-argument version returns the Wedderburn component (<Ref Sect="WedDec" />) of the finite group algebra <A>FG</A> realized by the strong Shoda pair (<A>K</A>,<A>H</A>) and the cyclotomic class <A>C</A> (or the cyclotomic class containing <A>C</A>). <P/> The versions ending in NC do not check if (<A>K</A>,<A>H</A>) is a strong Shoda pair of <M>G</M>. In the four-argument version it is also not checked whether <A>C</A> is either a generating <M>q</M>-cyclotomic class modulo the index of <A>H</A> in <A>K</A> or an integer coprime to the index of <A>H</A> in <A>K</A>. <Example> <![CDATA[ gap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;; gap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;; gap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);; gap> QG:=GroupRing( Rationals, G );; gap> FG:=GroupRing( GF(7), G );; gap> SimpleAlgebraByStrongSP( QG, K, H ); <crossed product over CF(16) of a group of size 2> gap> SimpleAlgebraByStrongSP( FG, K, H, [1,7] ); ( GF(7)^[ 2, 2 ] ) gap> SimpleAlgebraByStrongSP( FG, K, H, 1 ); ( GF(7)^[ 2, 2 ] ) ]]> </Example> </Description> </ManSection> <!-- ************************ --> <ManSection> <Oper Name="SimpleAlgebraByStrongSPInfo" Label="for rational group algebra" Arg="QG K H" Comm="The numerical data describing the output of SimpleAlgebraByStrongSP" /> <Oper Name="SimpleAlgebraByStrongSPInfoNC" Label="for rational group algebra" Arg="QG K H" Comm="The numerical data describing the output of SimpleAlgebraByStrongSPNC" /> <Oper Name="SimpleAlgebraByStrongSPInfo" Label="for semisimple finite group algebra" Arg="FG K H C" Comm="The numerical data describing the output of SimpleAlgebraByStrongSP" /> <Oper Name="SimpleAlgebraByStrongSPInfoNC" Label="for semisimple finite group algebra" Arg="FG K H C" Comm="The numerical data describing the output of SimpleAlgebraByStrongSPNC" /> <Returns> A numerical description of one simple algebra. </Returns> <Description> In the three-argument version the input must be formed by a <E>semisimple rational group algebra</E> (<Ref Sect="Semisimple" />) <A>QG</A> and two subgroups <A>K</A> and <A>H</A> of <M>G</M> which form a <E>strong Shoda pair</E> (<Ref Sect="SSPDef" />) of <M>G</M>. It returns the numerical information describing the Wedderburn component (<Ref Sect="NumDesc" />) of the rational group algebra <A>QG</A> realized by a the strong Shoda pair (<A>K</A>,<A>H</A>). <P/> In the four-argument version the first input is a semisimple finite group algebra <A>FG</A>, <A>(K,H)</A> is a strong Shoda pair of <M>G</M> and the fourth input data is either a generating <M>q</M>-cyclotomic class modulo the index of <A>H</A> in <A>K</A> or a representative of a generating <M>q</M>-cyclotomic class modulo the index of <A>H</A> in <A>K</A> (<Ref Sect="CyclotomicClass" />). It returns a pair of positive integers <M>[n,r]</M> which represent the <M>n\times n</M> matrix algebra over the field of order <M>r</M> which is isomorphic to the Wedderburn component of <A>FG</A> realized by a the strong Shoda pair (<A>K</A>,<A>H</A>) and the cyclotomic class <A>C</A> (or the cyclotomic class containing the integer <A>C</A>). <P/> The versions ending in NC do not check if (<A>K</A>,<A>H</A>) is a strong Shoda pair of <M>G</M>. In the four-argument version it is also not checked whether <A>C</A> is either a generating <M>q</M>-cyclotomic class modulo the index of <A>H</A> in <A>K</A> or an integer coprime with the index of <A>H</A> in <A>K</A>. <Example> <![CDATA[ gap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;; gap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;; gap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);; gap> QG:=GroupRing( Rationals, G );; gap> FG:=GroupRing( GF(7), G );; gap> SimpleAlgebraByStrongSP( QG, K, H ); <crossed product over CF(16) of a group of size 2> gap> SimpleAlgebraByStrongSPInfo( QG, K, H ); [ 1, NF(16,[ 1, 7 ]), 16, [ [ 2, 7, 8 ] ], [ ] ] gap> SimpleAlgebraByStrongSPInfo( FG, K, H, [1,7] ); [ 2, 7 ] gap> SimpleAlgebraByStrongSPInfo( FG, K, H, 1 ); [ 2, 7 ] ]]> </Example> </Description> </ManSection> </Section> </Chapter>