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<p><a id="X81DAF5267D30C83A" name="X81DAF5267D30C83A"></a></p>
<div class="ChapSects"><a href="chap3.html#X81DAF5267D30C83A">3 <span class="Heading">Strong Shoda pairs</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X807C74B07C4B99AF">3.1 <span class="Heading">Computing strong Shoda pairs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X820A398687A79B9D">3.1-1 StrongShodaPairs</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7B49C1BC834E57E3">3.2 <span class="Heading">Properties related with Shoda pairs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C17476F854F1E34">3.2-1 IsStrongShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X823B8DEC7ECC3326">3.2-2 IsShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80C4ED17809FC547">3.2-3 IsStronglyMonomial</a></span>
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<h3>3 <span class="Heading">Strong Shoda pairs</span></h3>
<p><a id="X807C74B07C4B99AF" name="X807C74B07C4B99AF"></a></p>
<h4>3.1 <span class="Heading">Computing strong Shoda pairs</span></h4>
<p><a id="X820A398687A79B9D" name="X820A398687A79B9D"></a></p>
<h5>3.1-1 StrongShodaPairs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> StrongShodaPairs</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>A list of pairs of subgroups of the input group.</p>
<p>The input should be a finite group <var class="Arg">G</var>.</p>
<p>Computes a list of representatives of the equivalence classes of <em>strong Shoda pairs</em> (<a href="chap7.html#X81DAF5267D30C83A"><b>7.15</b></a>) of a finite group <var class="Arg">G</var>.</p>
<table class="example">
<tr><td><pre>
gap> StrongShodaPairs( SymmetricGroup(4) );
[ [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ],
[ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]) ],
[ Group([ (1,2)(3,4), (1,3,2,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ])
],
[ Group([ (1,2)(3,4), (3,4), (1,3,2,4) ]), Group([ (1,2)(3,4), (3,4) ]) ],
[ Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ]
gap> StrongShodaPairs( DihedralGroup(64) );
[ [ <pc group of size 64 with 6 generators>,
Group([ f6, f5, f4, f3, f1, f2 ]) ],
[ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1*f2 ])
],
[ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f2 ]) ],
[ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1 ]) ],
[ Group([ f1*f2, f4*f5*f6, f5*f6, f6, f3, f3 ]),
Group([ f6, f5, f4, f1*f2 ]) ],
[ Group([ f6, f5, f2, f3, f4 ]), Group([ f6, f5 ]) ],
[ Group([ f6, f2, f3, f4, f5 ]), Group([ f6 ]) ],
[ Group([ f2, f3, f4, f5, f6 ]), Group([ ]) ] ]
</pre></td></tr></table>
<p><a id="X7B49C1BC834E57E3" name="X7B49C1BC834E57E3"></a></p>
<h4>3.2 <span class="Heading">Properties related with Shoda pairs</span></h4>
<p><a id="X7C17476F854F1E34" name="X7C17476F854F1E34"></a></p>
<h5>3.2-1 IsStrongShodaPair</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsStrongShodaPair</code>( <var class="Arg">G, K, H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The first argument should be a finite group <var class="Arg">G</var>, the second one a sugroup <var class="Arg">K</var> of <var class="Arg">G</var> and the third one a subgroup of <var class="Arg">K</var>.</p>
<p>Returns <code class="keyw">true</code> if (<var class="Arg">K</var>,<var class="Arg">H</var>) is a <em>strong Shoda pair</em> (<a href="chap7.html#X81DAF5267D30C83A"><b>7.15</b></a>) of <var class="Arg">G</var>, and <code class="keyw">false</code> otherwise.</p>
<table class="example">
<tr><td><pre>
gap> G:=SymmetricGroup(3);; K:=Group([(1,2,3)]);; H:=Group( () );;
gap> IsStrongShodaPair( G, K, H );
true
gap> IsStrongShodaPair( G, G, H );
false
gap> IsStrongShodaPair( G, K, K );
false
gap> IsStrongShodaPair( G, G, K );
true
</pre></td></tr></table>
<p><a id="X823B8DEC7ECC3326" name="X823B8DEC7ECC3326"></a></p>
<h5>3.2-2 IsShodaPair</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsShodaPair</code>( <var class="Arg">G, K, H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The first argument should be a finite group <var class="Arg">G</var>, the second a subgroup <var class="Arg">K</var> of <var class="Arg">G</var> and the third one a subgroup of <var class="Arg">K</var>.</p>
<p>Returns <code class="keyw">true</code> if (<var class="Arg">K</var>,<var class="Arg">H</var>) is a <em>Shoda pair</em> (<a href="chap7.html#X80C058BE81824B23"><b>7.14</b></a>) of <var class="Arg">G</var>.</p>
<p>Note that every strong Shoda pair is a Shoda pair, but the converse is not true.</p>
<table class="example">
<tr><td><pre>
gap> G:=AlternatingGroup(5);;
gap> K:=AlternatingGroup(4);;
gap> H := Group( (1,2)(3,4), (1,3)(2,4) );;
gap> IsStrongShodaPair( G, K, H );
false
gap> IsShodaPair( G, K, H );
true
</pre></td></tr></table>
<p><a id="X80C4ED17809FC547" name="X80C4ED17809FC547"></a></p>
<h5>3.2-3 IsStronglyMonomial</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsStronglyMonomial</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The input <var class="Arg">G</var> should be a finite group.</p>
<p>Returns <code class="keyw">true</code> if <var class="Arg">G</var> is a <em>strongly monomial</em> (<a href="chap7.html#X84C694978557EFE5"><b>7.16</b></a>) finite group.</p>
<table class="example">
<tr><td><pre>
gap> S4:=SymmetricGroup(4);;
gap> IsStronglyMonomial(S4);
true
gap> G:=SmallGroup(24,3);;
gap> IsStronglyMonomial(G);
false
gap> IsMonomial(G);
false
gap> G:=SmallGroup(1000,86);;
gap> IsMonomial(G);
true
gap> IsStronglyMonomial(G);
false
</pre></td></tr></table>
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