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<p><a id="X80C058BE81824B23" name="X80C058BE81824B23"></a></p>
<div class="ChapSects"><a href="chap3.html#X80C058BE81824B23">3 <span class="Heading">Shoda pairs</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8072BA2B87199557">3.1 <span class="Heading">Computing extremely strong Shoda pairs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86B2AFF87D26FC75">3.1-1 ExtremelyStrongShodaPairs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X807C74B07C4B99AF">3.2 <span class="Heading">Computing strong Shoda pairs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X820A398687A79B9D">3.2-1 StrongShodaPairs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B49C1BC834E57E3">3.3 <span class="Heading">Properties related with Shoda pairs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A851A00809B4C92">3.3-1 IsExtremelyStrongShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C17476F854F1E34">3.3-2 IsStrongShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X823B8DEC7ECC3326">3.3-3 IsShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80C4ED17809FC547">3.3-4 IsStronglyMonomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8485C39787CF0797">3.3-5 IsNormallyMonomial</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Shoda pairs</span></h3>

<p><a id="X8072BA2B87199557" name="X8072BA2B87199557"></a></p>

<h4>3.1 <span class="Heading">Computing extremely strong Shoda pairs</span></h4>

<p><a id="X86B2AFF87D26FC75" name="X86B2AFF87D26FC75"></a></p>

<h5>3.1-1 ExtremelyStrongShodaPairs</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtremelyStrongShodaPairs</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: 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>extremely strong Shoda pairs</em> (<a href="chap9.html#X81B5CE0378DC4913"><span class="RefLink">9.16</span></a>) of a finite group <var class="Arg">G</var>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ExtremelyStrongShodaPairs(DihedralGroup(32));</span>
[ [ <pc group of size 32 with 5 generators>, 
      <pc group of size 32 with 5 generators> ], 
  [ <pc group of size 32 with 5 generators>, 
      Group([ f1*f2*f3*f4*f5, f3, f4, f5 ]) ], 
  [ <pc group of size 32 with 5 generators>, Group([ f2, f3, f4, f5 ]) ], 
  [ <pc group of size 32 with 5 generators>, Group([ f1, f3, f4, f5 ]) ], 
  [ Group([ f1*f2*f3*f4*f5, f3, f4, f5 ]), Group([ f1*f2*f4*f5, f4, f5 ]) ], 
  [ Group([ f2, f3, f4, f5 ]), Group([ f5 ]) ], 
  [ Group([ f2, f3, f4, f5 ]), Group([  ]) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtremelyStrongShodaPairs(SL(2,3));          </span>
[ [ SL(2,3), SL(2,3) ], 
  [ SL(2,3), 
      Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], 
          [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], 
          [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ], 
  [ 
      Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], 
          [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], 
          [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]), 
      Group([ [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ], 
          [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtremelyStrongShodaPairs(SymmetricGroup(5));</span>
[ [ Sym( [ 1 .. 5 ] ), Sym( [ 1 .. 5 ] ) ], 
  [ Sym( [ 1 .. 5 ] ), Alt( [ 1 .. 5 ] ) ] ]

</pre></div>

<p><a id="X807C74B07C4B99AF" name="X807C74B07C4B99AF"></a></p>

<h4>3.2 <span class="Heading">Computing strong Shoda pairs</span></h4>

<p><a id="X820A398687A79B9D" name="X820A398687A79B9D"></a></p>

<h5>3.2-1 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>Returns: 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="chap9.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of a finite group <var class="Arg">G</var>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ssp:=StrongShodaPairs( SymmetricGroup(4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(ssp);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">List(ssp,x->List(x,StructureDescription));</span>
[ [ "S4", "S4" ], [ "S4", "A4" ], [ "A4", "C2 x C2" ], [ "D8", "C2 x C2" ], 
  [ "D8", "C4" ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ssp:=StrongShodaPairs( DihedralGroup(64) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(ssp);</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">List(ssp,x->List(x,StructureDescription));</span>
[ [ "D64", "D64" ], [ "D64", "D32" ], [ "D64", "C32" ], [ "D64", "D32" ], 
  [ "D32", "D16" ], [ "C32", "C4" ], [ "C32", "C2" ], [ "C32", "1" ] ]

</pre></div>

<p><a id="X7B49C1BC834E57E3" name="X7B49C1BC834E57E3"></a></p>

<h4>3.3 <span class="Heading">Properties related with Shoda pairs</span></h4>

<p><a id="X7A851A00809B4C92" name="X7A851A00809B4C92"></a></p>

<h5>3.3-1 IsExtremelyStrongShodaPair</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsExtremelyStrongShodaPair</code>( <var class="Arg">G</var>, <var class="Arg">K</var>, <var class="Arg">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 normal 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 an <em>extremely strong Shoda pair</em> (<a href="chap9.html#X81B5CE0378DC4913"><span class="RefLink">9.16</span></a>) of <var class="Arg">G</var>, and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(4);; K:=Group( (1,3,2,4), (3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H1:=Group( (2,4,3), (1,4)(2,3), (1,3)(2,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H2:=Group( (3,4), (1,2)(3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExtremelyStrongShodaPair( G, G, H1 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExtremelyStrongShodaPair( G, K, H2 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExtremelyStrongShodaPair( G, G, H2 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExtremelyStrongShodaPair( G, G, K );</span>
false

</pre></div>

<p><a id="X7C17476F854F1E34" name="X7C17476F854F1E34"></a></p>

<h5>3.3-2 IsStrongShodaPair</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsStrongShodaPair</code>( <var class="Arg">G</var>, <var class="Arg">K</var>, <var class="Arg">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="chap9.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <var class="Arg">G</var>, and <code class="keyw">false</code> otherwise.</p>

<p>Note that every extremely strong Shoda pair is a strong Shoda pair, but the converse is not true.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(4);; K:=Group( (1,3,2,4), (3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H1:=Group( (2,4,3), (1,4)(2,3), (1,3)(2,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H2:=Group( (3,4), (1,2)(3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStrongShodaPair( G, G, H1 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExtremelyStrongShodaPair( G, K, H2 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStrongShodaPair( G, K, H2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStrongShodaPair( G, G, K );</span>
false

</pre></div>

<p><a id="X823B8DEC7ECC3326" name="X823B8DEC7ECC3326"></a></p>

<h5>3.3-3 IsShodaPair</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsShodaPair</code>( <var class="Arg">G</var>, <var class="Arg">K</var>, <var class="Arg">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="chap9.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></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>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=AlternatingGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Group( (1,2)(3,4), (1,3)(2,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStrongShodaPair( G, K, H );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsShodaPair( G, K, H );</span>
true

</pre></div>

<p><a id="X80C4ED17809FC547" name="X80C4ED17809FC547"></a></p>

<h5>3.3-4 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="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>) finite group.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">S4:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStronglyMonomial(S4);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(24,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStronglyMonomial(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonomial(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(1000,86);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonomial(G);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStronglyMonomial(G);</span>
false

</pre></div>

<p><a id="X8485C39787CF0797" name="X8485C39787CF0797"></a></p>

<h5>3.3-5 IsNormallyMonomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNormallyMonomial</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 finite <em>normally monomial</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>) group.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput"> D24:=DihedralGroup(24);</span>
<pc group of size 24 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormallyMonomial(D24);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(192,1023);</span>
<pc group of size 192 with 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormallyMonomial(G);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(1029,12); </span>
<pc group of size 1029 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormallyMonomial(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStronglyMonomial(G);  </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SL(2,3);            </span>
SL(2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormallyMonomial(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsStronglyMonomial(G);</span>
false

</pre></div>


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