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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap5_mj.html">[Previous Chapter]</a>    <a href="chap7_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap6.html">[MathJax off]</a></p>
<p><a id="X7D3C0B1F7A66056F" name="X7D3C0B1F7A66056F"></a></p>
<div class="ChapSects"><a href="chap6_mj.html#X7D3C0B1F7A66056F">6 <span class="Heading">Useful properties and functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7BA5D68A86B8C772">6.1 <span class="Heading">Semisimple group algebras of finite groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7EF856E880722311">6.1-1 IsSemisimpleZeroCharacteristicGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X85999B6A7C52E305">6.1-2 IsSemisimpleRationalGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X79289F7F7FC04846">6.1-3 IsSemisimpleANFGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7B546E2D7FB561BA">6.1-4 IsSemisimpleFiniteGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8337F25387C53B02">6.1-5 IsTwistingTrivial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X86121BD77F7E5C7A">6.2 <span class="Heading">Operations with group rings elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A2BF4527E08803C">6.2-1 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7FE417DD837987B4">6.2-2 OnPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X798CEA1F80D355EE">6.2-3 AverageSum</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7AAB3882785C04E0">6.3 <span class="Heading">Cyclotomic classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7D7BDF5087C8F4C6">6.3-1 CyclotomicClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7FA101AE7BC33671">6.3-2 IsCyclotomicClass</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7B16423A7FBED034">6.4 <span class="Heading">Other commands</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X872510997A7AF31D">6.4-1 InfoWedderga</a></span>
</div></div>
</div>

<h3>6 <span class="Heading">Useful properties and functions</span></h3>

<p><a id="X7BA5D68A86B8C772" name="X7BA5D68A86B8C772"></a></p>

<h4>6.1 <span class="Heading">Semisimple group algebras of finite groups</span></h4>

<p><a id="X7EF856E880722311" name="X7EF856E880722311"></a></p>

<h5>6.1-1 IsSemisimpleZeroCharacteristicGroupAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemisimpleZeroCharacteristicGroupAlgebra</code>( <var class="Arg">KG</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The input must be a group ring.</p>

<p>Returns <code class="keyw">true</code> if the input <var class="Arg">KG</var> is a <em>semisimple group algebra</em> (<a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) over a field of characteristic zero (that is if <span class="SimpleMath">\(G\)</span> is finite), and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">CG:=GroupRing( GaussianRationals, DihedralGroup(16) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleZeroCharacteristicGroupAlgebra( CG );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing( GF(2), SymmetricGroup(3) );;                    </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleZeroCharacteristicGroupAlgebra( FG );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup("a");</span>
<free group on the generators [ a ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qf:=GroupRing(Rationals,f);</span>
<algebra-with-one over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleZeroCharacteristicGroupAlgebra(Qf);</span>
false

</pre></div>

<p><a id="X85999B6A7C52E305" name="X85999B6A7C52E305"></a></p>

<h5>6.1-2 IsSemisimpleRationalGroupAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemisimpleRationalGroupAlgebra</code>( <var class="Arg">KG</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The input must be a group ring.</p>

<p>Returns <code class="keyw">true</code> if <var class="Arg">KG</var> is a <em>semisimple rational group algebra</em> (<a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing( Rationals, SymmetricGroup(4) );;       </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleRationalGroupAlgebra( QG );       </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CG:=GroupRing( GaussianRationals, DihedralGroup(16) );;               </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleRationalGroupAlgebra( CG );                              </span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing( GF(2), SymmetricGroup(3) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleRationalGroupAlgebra( FG );</span>
false

</pre></div>

<p><a id="X79289F7F7FC04846" name="X79289F7F7FC04846"></a></p>

<h5>6.1-3 IsSemisimpleANFGroupAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemisimpleANFGroupAlgebra</code>( <var class="Arg">KG</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The input must be a group ring.</p>

<p>Returns <code class="keyw">true</code> if <var class="Arg">KG</var> is the group algebra of a finite group over a subfield of a cyclotomic extension of the rationals and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleANFGroupAlgebra( GroupRing( NF(5,[4]) , CyclicGroup(28) ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleANFGroupAlgebra( GroupRing( GF(11) , CyclicGroup(28) ) );</span>
false

</pre></div>

<p><a id="X7B546E2D7FB561BA" name="X7B546E2D7FB561BA"></a></p>

<h5>6.1-4 IsSemisimpleFiniteGroupAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemisimpleFiniteGroupAlgebra</code>( <var class="Arg">KG</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The input must be a group ring.</p>

<p>Returns <code class="keyw">true</code> if <var class="Arg">KG</var> is a <em>semisimple finite group algebra</em> (<a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>), that is a group algebra of a finite group <span class="SimpleMath">\(G\)</span> over a field <span class="SimpleMath">\(K\)</span> of order coprime to the order of <span class="SimpleMath">\(G\)</span>, and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing( GF(5), SymmetricGroup(3) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleFiniteGroupAlgebra( FG );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">KG:=GroupRing( GF(2), SymmetricGroup(3) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleFiniteGroupAlgebra( KG ); </span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing( Rationals, SymmetricGroup(4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemisimpleFiniteGroupAlgebra( QG );</span>
false

</pre></div>

<p><a id="X8337F25387C53B02" name="X8337F25387C53B02"></a></p>

<h5>6.1-5 IsTwistingTrivial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTwistingTrivial</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">K</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The input must be a group and a strong Shoda pair of the group.</p>

<p>Returns <code class="keyw">true</code> if the simple algebra <span class="SimpleMath">\(ℚGe(G,H,K)\)</span> has a <em>trivial twisting</em> (<a href="chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>), and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=StrongShodaPairs(G)[5][1];</span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=StrongShodaPairs(G)[5][2]; </span>
Group([ f1*f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTwistingTrivial(G,H,K);</span>
true

</pre></div>

<p><a id="X86121BD77F7E5C7A" name="X86121BD77F7E5C7A"></a></p>

<h4>6.2 <span class="Heading">Operations with group rings elements</span></h4>

<p><a id="X7A2BF4527E08803C" name="X7A2BF4527E08803C"></a></p>

<h5>6.2-1 Centralizer</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centralizer</code>( <var class="Arg">G</var>, <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A subgroup of a group <var class="Arg">G</var>.</p>

<p>The input should be formed by a finite group <var class="Arg">G</var> and an element <var class="Arg">x</var> of a group ring <span class="SimpleMath">\(FH\)</span> whose underlying group <span class="SimpleMath">\(H\)</span> contains <var class="Arg">G</var> as a subgroup.</p>

<p>Returns the centralizer of <var class="Arg">x</var> in <var class="Arg">G</var>.</p>

<p>This operation adds a new method to the operation that already exists in <strong class="pkg">GAP</strong>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">D16 := DihedralGroup(16);</span>
<pc group of size 16 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">QD16 := GroupRing( Rationals, D16 );</span>
<algebra-with-one over Rationals, with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=QD16.1;b:=QD16.2;</span>
(1)*f1
(1)*f2
<span class="GAPprompt">gap></span> <span class="GAPinput">e := PrimitiveCentralIdempotentsByStrongSP( QD16)[3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Centralizer( D16, a);</span>
Group([ f1, f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Centralizer( D16, b);</span>
Group([ f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Centralizer( D16, a+b);</span>
Group([ f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Centralizer( D16, e);</span>
Group([ f1, f2 ])

</pre></div>

<p><a id="X7FE417DD837987B4" name="X7FE417DD837987B4"></a></p>

<h5>6.2-2 OnPoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnPoints</code>( <var class="Arg">x</var>, <var class="Arg">g</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">‣ \^</code>( <var class="Arg">x</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: An element of a group ring.</p>

<p>The input should be formed by an element <var class="Arg">x</var> of a group ring <span class="SimpleMath">\(FG\)</span> and an element <var class="Arg">g</var> in the underlying group <span class="SimpleMath">\(G\)</span> of <span class="SimpleMath">\(FG\)</span>.</p>

<p>Returns the conjugate <span class="SimpleMath">\(x^g = g^{-1} x g\)</span> of <var class="Arg">x</var> by <var class="Arg">g</var>. Usage of <code class="code">x^g</code> produces the same output.</p>

<p>This operation adds a new method to the operation that already exists in <strong class="pkg">GAP</strong>.</p>

<p>The following example is a continuation of the example from the description of <code class="func">Centralizer</code> (<a href="chap6_mj.html#X7A2BF4527E08803C"><span class="RefLink">6.2-1</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(D16,x->a^x=a);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(D16,x->e^x=e);</span>
true

</pre></div>

<p><a id="X798CEA1F80D355EE" name="X798CEA1F80D355EE"></a></p>

<h5>6.2-3 AverageSum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AverageSum</code>( <var class="Arg">RG</var>, <var class="Arg">X</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: An element of a group ring.</p>

<p>The input must be composed of a group ring <var class="Arg">RG</var> and a finite subset <var class="Arg">X</var> of the underlying group <span class="SimpleMath">\(G\)</span> of <var class="Arg">RG</var>. The order of <var class="Arg">X</var> must be invertible in the coefficient ring <span class="SimpleMath">\(R\)</span> of <var class="Arg">RG</var>.</p>

<p>Returns the element of the group ring <var class="Arg">RG</var> that is equal to the sum of all elements of <var class="Arg">X</var> divided by the order of <var class="Arg">X</var>.</p>

<p>If <var class="Arg">X</var> is a subgroup of <span class="SimpleMath">\(G\)</span> then the output is an idempotent of <span class="SimpleMath">\(RG\)</span> which is central if and only if <var class="Arg">X</var> is normal in <span class="SimpleMath">\(G\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(16);;               </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing( Rationals, G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing( GF(5), G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=AverageSum( QG, DerivedSubgroup(G) );</span>
(1/4)*<identity> of ...+(1/4)*f3+(1/4)*f4+(1/4)*f3*f4
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=AverageSum( FG, DerivedSubgroup(G) ); </span>
(Z(5)^2)*<identity> of ...+(Z(5)^2)*f3+(Z(5)^2)*f4+(Z(5)^2)*f3*f4
<span class="GAPprompt">gap></span> <span class="GAPinput">G=Centralizer(G,e);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=Subgroup(G,[G.1]);</span>
Group([ f1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=AverageSum( QG, H );</span>
(1/2)*<identity> of ...+(1/2)*f1
<span class="GAPprompt">gap></span> <span class="GAPinput">G=Centralizer(G,e);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal(G,H);</span>
false

</pre></div>

<p><a id="X7AAB3882785C04E0" name="X7AAB3882785C04E0"></a></p>

<h4>6.3 <span class="Heading">Cyclotomic classes</span></h4>

<p><a id="X7D7BDF5087C8F4C6" name="X7D7BDF5087C8F4C6"></a></p>

<h5>6.3-1 CyclotomicClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclotomicClasses</code>( <var class="Arg">q</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A partition of <span class="SimpleMath">\([ 0 .. n ]\)</span>.</p>

<p>The input should be formed by two relatively prime positive integers.</p>

<p>Returns the list <var class="Arg">q</var>-<em>cyclotomic classes </em> (<a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>) modulo <var class="Arg">n</var>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicClasses( 2, 21 );</span>
[ [ 0 ], [ 1, 2, 4, 8, 16, 11 ], [ 3, 6, 12 ], [ 5, 10, 20, 19, 17, 13 ],
  [ 7, 14 ], [ 9, 18, 15 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicClasses( 10, 21 );</span>
[ [ 0 ], [ 1, 10, 16, 13, 4, 19 ], [ 2, 20, 11, 5, 8, 17 ],
  [ 3, 9, 6, 18, 12, 15 ], [ 7 ], [ 14 ] ]

</pre></div>

<p><a id="X7FA101AE7BC33671" name="X7FA101AE7BC33671"></a></p>

<h5>6.3-2 IsCyclotomicClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclotomicClass</code>( <var class="Arg">q</var>, <var class="Arg">n</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The input should be formed by two relatively prime positive integers <var class="Arg">q</var> and <var class="Arg">n</var> and a sublist <var class="Arg">C</var> of <span class="SimpleMath">\([ 0 .. n ]\)</span>.</p>

<p>Returns <code class="keyw">true</code> if <var class="Arg">C</var> is a <var class="Arg">q</var>-<em>cyclotomic class</em> (<a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>) modulo <var class="Arg">n</var> and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicClass( 2, 7, [1,2,4] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicClass( 2, 21, [1,2,4] );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicClass( 2, 21, [3,6,12] );</span>
true

</pre></div>

<p><a id="X7B16423A7FBED034" name="X7B16423A7FBED034"></a></p>

<h4>6.4 <span class="Heading">Other commands</span></h4>

<p><a id="X872510997A7AF31D" name="X872510997A7AF31D"></a></p>

<h5>6.4-1 InfoWedderga</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoWedderga</code></td><td class="tdright">( info class )</td></tr></table></div>
<p><code class="code">InfoWedderga</code> is a special Info class for <strong class="pkg">Wedderga</strong> algorithms. It has 3 levels: 0, 1 (default) and 2. To change the info level to <code class="code">k</code>, use the command <code class="code">SetInfoLevel(InfoWedderga, k)</code>.</p>

<p>In the example below we use this mechanism to see more details about the Wedderburn components each time when we call <code class="code">WedderburnDecomposition</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel(InfoWedderga, 2);   </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( GroupRing( CF(5), DihedralGroup( 16 ) ) );</span>
#I  Info version : [ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], 
  [ 1, CF(5) ], [ 2, CF(5) ], [ 2, NF(40,[ 1, 31 ]) ] ]
[ 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>


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