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<a id="X87273420791F220E" name="X87273420791F220E"></a>
<div class="ChapSects"><a href="chap2.html#X87273420791F220E">2 Wedderburn decomposition</a>
<div class="ContSect"> <a href="chap2.html#X7C902C667D137851">2.1 Wedderburn decomposition of a group algebra</a>

<div class="ContSSBlock">
 <a href="chap2.html#X7F1779ED8777F3E7">2.1-1 WedderburnDecomposition</a>
 <a href="chap2.html#X8710F98A85F0DD29">2.1-2 WedderburnDecompositionInfo</a>
</div></div>
<div class="ContSect"> <a href="chap2.html#X7D06959F7D444C55">2.2 Simple quotients</a>

<div class="ContSSBlock">
 <a href="chap2.html#X8349114C83161C2D">2.2-1 SimpleAlgebraByCharacter</a>
 <a href="chap2.html#X876FD2367E64462D">2.2-2 SimpleAlgebraByCharacterInfo</a>
 <a href="chap2.html#X812D667D7D913EB5">2.2-3 SimpleAlgebraByStrongSP</a>
 <a href="chap2.html#X858152C882129A0B">2.2-4 SimpleAlgebraByStrongSPInfo</a>
</div></div>
</div>

<h3>2 Wedderburn decomposition</h3>

<a id="X7C902C667D137851" name="X7C902C667D137851"></a>

<h4>2.1 Wedderburn decomposition of a group algebra</h4>

<a id="X7F1779ED8777F3E7" name="X7F1779ED8777F3E7"></a>

<h5>2.1-1 WedderburnDecomposition</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WedderburnDecomposition</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A list of simple algebras.

The input <var class="Arg">FG</var> 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 Wedderburn components (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) of the group algebra <var class="Arg">FG</var>. If F is an abelian number field then each Wedderburn component is given as a matrix algebra of a cyclotomic algebra (<a href="chap9.html#X8099A8C784255672">9.11</a>). If F is a finite field then the Wedderburn components are given as matrix algebras over finite fields.

<div class="example"><pre>

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> ]

</pre></div>

The previous examples show that if D_16 denotes the dihedral group of order 16 then the Wedderburn decomposition (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) of F_5 D_16, ℚ D_16 and ℚ (ξ_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(ξ_8)/K,t) is a cyclotomic algebra (<a href="chap9.html#X8099A8C784255672">9.11</a>) with the centre K=NF(8,[ 1, 7 ])= ℚ (sqrt2), (F(ξ_40)/F,t) = ℚ (sqrt2,ξ_5) is a cyclotomic algebra with centre F=NF(40,[ 1, 31 ]) and ξ_n denotes a n-th root of unity.

Two more examples:

<div class="example"><pre>

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 ] ) ]

</pre></div>

In some cases, in characteristic zero, some entries of the output of <code class="func">WedderburnDecomposition</code> do not provide full matrix algebras over a cyclotomic algebra (<a href="chap9.html#X8099A8C784255672">9.11</a>), 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 <a href="chap9.html#X84BB4A6081EAE905">9.3</a> for a theoretical explanation of this phenomenon. In this case a warning message is displayed.

<div class="example"><pre>

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> ] ]

</pre></div>

<a id="X8710F98A85F0DD29" name="X8710F98A85F0DD29"></a>

<h5>2.1-2 WedderburnDecompositionInfo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WedderburnDecompositionInfo</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A list with each entry a numerical description of a cyclotomic algebra (<a href="chap9.html#X8099A8C784255672">9.11</a>).

The input <var class="Arg">FG</var> 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 <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7">2.1-1</a>).

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 n× 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,α,β are non-negative integers, satisfying the conditions mentioned in Section <a href="chap9.html#X84A142407B7565E0">9.12</a>. It represents the n× 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 ξ_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,α_i,β_i,γ_i,j are non-negative integers. It represents the n× n matrix algebra M_n(A) over the cyclotomic algebra (<a href="chap9.html#X8099A8C784255672">9.11</a>)


 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 ξ_k is a primitive k-th root of unity (see <a href="chap9.html#X84A142407B7565E0">9.12</a>).

<div class="example"><pre>

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 ]) ] ]

</pre></div>

The interpretation of the previous example gives rise to the following Wedderburn decompositions (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>), where D_16 is the dihedral group of order 16 and ξ_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})).
 

<div class="example"><pre>

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 ] ]

</pre></div>

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 <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7">2.1-1</a>) and <code class="func">WedderburnDecompositionInfo</code>.

The output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7">2.1-1</a>) 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 crossed products (<a href="chap9.html#X7FB21779832CE1CB">9.6</a>) with coefficients in the cyclotomic fields ℚ (ξ_8) and ℚ (ξ_3) respectively. This output can be used as a GAP object, but it does not give clear information on the structure of the algebras A and B.

The numerical information displayed by <code class="func">WedderburnDecompositionInfo</code> 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 (ξ+ξ^-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. <a href="chapBib.html#biBR">[Rei03]</a>).

The next example shows the output of <code class="code">WedderburnDecompositionInfo</code> for ℚ G and ℚ (ξ_3) G, where G=SmallGroup(48,15). The user can compare it with the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7">2.1-1</a>) for the same group in the previous section. Notice that the last entry of the Wedderburn decomposition (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) of ℚ G is not given as a matrix algebra of a cyclic algebra. However, the corresponding entry of ℚ (ξ_3) G is a matrix algebra of a cyclic algebra.

<div class="example"><pre>

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) ] ]

</pre></div>

In some cases some of the first entries of the output of <code class="func">WedderburnDecompositionInfo</code> are not integers and so the correspoding Wedderburn components (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) are given as "fractional matrix algebras" of cyclotomic algebras (<a href="chap9.html#X8099A8C784255672">9.11</a>). See <a href="chap9.html#X84BB4A6081EAE905">9.3</a> for a theoretical explanation of this phenomenon. In that case a warning message will be displayed during the first call of <code class="code">WedderburnDecompositionInfo</code>.

<div class="example"><pre>

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 ] ] ]

</pre></div>

The interpretation of the output in the previous example gives rise to the following Wedderburn decomposition (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) 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 Brauer equivalent (<a href="chap9.html#X7A24D5407F72C633">9.5</a>) 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 Brauer equivalent (<a href="chap9.html#X7A24D5407F72C633">9.5</a>) 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.

<a id="X7D06959F7D444C55" name="X7D06959F7D444C55"></a>

<h4>2.2 Simple quotients</h4>

<a id="X8349114C83161C2D" name="X8349114C83161C2D"></a>

<h5>2.2-1 SimpleAlgebraByCharacter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByCharacter</code>( <var class="Arg">FG</var>, <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A simple algebra.

The first input <var class="Arg">FG</var> should be a semisimple group algebra (<a href="chap9.html#X7FDD93FB79ADCC91">9.2</a>) over a finite group G and the second input should be an irreducible character of G.

The output is a matrix algebra of a cyclotomic algebras (<a href="chap9.html#X8099A8C784255672">9.11</a>) which is isomorphic to the unique Wedderburn component (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) A of <var class="Arg">FG</var> such that χ(A)ne 0.

<div class="example"><pre>

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 ] ) ]

</pre></div>

<a id="X876FD2367E64462D" name="X876FD2367E64462D"></a>

<h5>2.2-2 SimpleAlgebraByCharacterInfo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByCharacterInfo</code>( <var class="Arg">FG</var>, <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: The numerical description of the output of <code class="func">SimpleAlgebraByCharacter</code> (<a href="chap2.html#X8349114C83161C2D">2.2-1</a>).

The first input <var class="Arg">FG</var> is a semisimple group algebra (<a href="chap9.html#X7FDD93FB79ADCC91">9.2</a>) over a finite group G and the second input is an irreducible character of G.

The output is the numerical description <a href="chap9.html#X84A142407B7565E0">9.12</a> of the cyclotomic algebra (<a href="chap9.html#X8099A8C784255672">9.11</a>) which is isomorphic to the unique Wedderburn component (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) A of <var class="Arg">FG</var> such that χ(A)ne 0.

See <a href="chap9.html#X84A142407B7565E0">9.12</a> for the interpretation of the numerical information given by the output.

<div class="example"><pre>

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 ]) ] ]

</pre></div>

<a id="X812D667D7D913EB5" name="X812D667D7D913EB5"></a>

<h5>2.2-3 SimpleAlgebraByStrongSP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSP</code>( <var class="Arg">QG</var>, <var class="Arg">K</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSPNC</code>( <var class="Arg">QG</var>, <var class="Arg">K</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSP</code>( <var class="Arg">FG</var>, <var class="Arg">K</var>, <var class="Arg">H</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSPNC</code>( <var class="Arg">FG</var>, <var class="Arg">K</var>, <var class="Arg">H</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A simple algebra.

In the three-argument version the input must be formed by a semisimple rational group algebra <var class="Arg">QG</var> (see <a href="chap9.html#X7FDD93FB79ADCC91">9.2</a>) and two subgroups <var class="Arg">K</var> and <var class="Arg">H</var> of G which form a strong Shoda pair (<a href="chap9.html#X7E3479527BAE5B9E">9.15</a>) of G.

The three-argument version returns the Wedderburn component (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) of the rational group algebra <var class="Arg">QG</var> realized by the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>).

In the four-argument version the first argument is a semisimple finite group algebra <var class="Arg">FG</var>, <var class="Arg">(K,H)</var> is a strong Shoda pair of G and the fourth input data is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or a representative of a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> (see <a href="chap9.html#X800D8C5087D79DC8">9.19</a>).

The four-argument version returns the Wedderburn component (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) of the finite group algebra <var class="Arg">FG</var> realized by the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>) and the cyclotomic class <var class="Arg">C</var> (or the cyclotomic class containing <var class="Arg">C</var>).

The versions ending in NC do not check if (<var class="Arg">K</var>,<var class="Arg">H</var>) is a strong Shoda pair of G. In the four-argument version it is also not checked whether <var class="Arg">C</var> is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or an integer coprime to the index of <var class="Arg">H</var> in <var class="Arg">K</var>.

<div class="example"><pre>

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 ] )

</pre></div>

<a id="X858152C882129A0B" name="X858152C882129A0B"></a>

<h5>2.2-4 SimpleAlgebraByStrongSPInfo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSPInfo</code>( <var class="Arg">QG</var>, <var class="Arg">K</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSPInfoNC</code>( <var class="Arg">QG</var>, <var class="Arg">K</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSPInfo</code>( <var class="Arg">FG</var>, <var class="Arg">K</var>, <var class="Arg">H</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleAlgebraByStrongSPInfoNC</code>( <var class="Arg">FG</var>, <var class="Arg">K</var>, <var class="Arg">H</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A numerical description of one simple algebra.

In the three-argument version the input must be formed by a semisimple rational group algebra (<a href="chap9.html#X7FDD93FB79ADCC91">9.2</a>) <var class="Arg">QG</var> and two subgroups <var class="Arg">K</var> and <var class="Arg">H</var> of G which form a strong Shoda pair (<a href="chap9.html#X7E3479527BAE5B9E">9.15</a>) of G. It returns the numerical information describing the Wedderburn component (<a href="chap9.html#X84A142407B7565E0">9.12</a>) of the rational group algebra <var class="Arg">QG</var> realized by a the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>).

In the four-argument version the first input is a semisimple finite group algebra <var class="Arg">FG</var>, <var class="Arg">(K,H)</var> is a strong Shoda pair of G and the fourth input data is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or a representative of a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> (<a href="chap9.html#X800D8C5087D79DC8">9.19</a>). It returns a pair of positive integers [n,r] which represent the n× n matrix algebra over the field of order r which is isomorphic to the Wedderburn component of <var class="Arg">FG</var> realized by a the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>) and the cyclotomic class <var class="Arg">C</var> (or the cyclotomic class containing the integer <var class="Arg">C</var>).

The versions ending in NC do not check if (<var class="Arg">K</var>,<var class="Arg">H</var>) is a strong Shoda pair of G. In the four-argument version it is also not checked whether <var class="Arg">C</var> is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or an integer coprime with the index of <var class="Arg">H</var> in <var class="Arg">K</var>.

<div class="example"><pre>

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 ]

</pre></div>

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