Quelle decomp.xml Sprache: XML

<Chapter Label="Decomp">
<Heading>Wedderburn decomposition</Heading>

<Section Label="DecompDecomp">
<Heading>Wedderburn decomposition of a group algebra</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>.
 

 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.


Two more examples:

<Example>
<![CDATA[
gap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );
[ Rationals, Rationals, Rationals, Rationals,
 <crossed product with center Rationals over CF(3) of a group of size 2>,
 <crossed product with center Rationals over GaussianRationals of a group of \
size 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>, ( CF(3)^[ 2, 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 ] ),
 <crossed product with center CF(3) over AsField( CF(3), CF(
 12) ) of a group of size 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>, ( CF(3)^[ 2, 2 ] ),
 ( CF(3)^[ 2, 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>.
 

 This function is a numerical counterpart of
 <Ref Attr="WedderburnDecomposition" />.
 

 It returns a list formed by lists of lengths 2, 4 or 5. 

 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>.

 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. 

 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 ], [ 2, NF(8,[ 1, 7 ]) ] ]
gap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) );
[ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ],
 [ 2, NF(40,[ 1, 31 ]) ] ]
]]>
</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, <crossed product with center Rationals over CF(
 3) of a group of size 2>, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ) ]
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 ], [ 2, Rationals ], [ 3, Rationals ],
 [ 3, Rationals ] ]
]]>
</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" />. 

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>. 

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. 

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" />).


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 ], [ 2, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ],
 [ 2, CF(3) ], [ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ] ]
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) ], [ 2, CF(3) ], [ 2, NF(24,[ 1, 7 ]) ], [ 2, CF(3) ],
 [ 2, CF(3) ], [ 4, CF(3) ] ]
]]>
</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>.

 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> chi := First(Irr( A5 ), chi -> Degree(chi) = 3);;
gap> SimpleAlgebraByCharacter( GroupRing( Rationals, A5 ), chi );
( NF(5,[ 1, 4 ])^[ 3, 3 ] )
gap> SimpleAlgebraByCharacter( GroupRing( GF(7), A5 ), chi );
( GF(7^2)^[ 3, 3 ] )
gap> G:=SmallGroup(128,100);
<pc group of size 128 with 7 generators>
gap> chi4:=Filtered(Irr(G),x->Degree(x)=4);;
gap> List(chi4,x->SimpleAlgebraByCharacter(GroupRing(Rationals,G),x));
[ ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
 [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
 ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
 [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
 ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
 [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
 ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
 [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 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>.
 

 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>. 

 See <Ref Sect="NumDesc" /> for the interpretation of the numerical
 information given by the output.

<Alt Only="LaTeX">\newpage</Alt>
<Example>
<![CDATA[
gap> G:=SmallGroup(128,100);
<pc group of size 128 with 7 generators>
gap> QG:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 7 generators>
gap> chi4:=Filtered(Irr(G),x->Degree(x)=4);;
gap> List(chi4,x->SimpleAlgebraByCharacterInfo(QG,x));
[ [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ],
 [ 4, NF(8,[ 1, 3 ]) ] ]
]]>
</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>. 

 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>).
 

 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" />). 

 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>). 

 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>). 

 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>). 

 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>

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