<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
<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="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="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="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="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="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="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 outputis 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="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="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> ]
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