<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (Wedderga) - Chapter 3: Strong Shoda pairs</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chap4.html">Next Chapter</a> </div> <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> </div> <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> </div> </div> <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> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chap4.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>