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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html"

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<p id="mathjaxlink" class="pcenter"><a href="chap3_mj.html">[MathJax on]</a></p>
<p><a id="X7B87CECD83EA77DF" name="X7B87CECD83EA77DF"></a></p>
<div class="ChapSects"><a href="chap3.html#X7B87CECD83EA77DF">3 <span class="Heading">Permutability of Subgroups in Finite Groups</span></a>
<="ContSect">< class"ocline"><pan class=nocss>&bsp;span<a href="chap3.tml#834D77E17960C0B7"3. span class"eading"> functionsspan>/java.lang.StringIndexOutOfBoundsException: Index 176 out of bounds for length 176
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X862727A87C26B10B">3.1-1 ArePermutableSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X875BEB9B7D0DBC9A">3.2 <span class="Heading">Embedding properties related to permutability</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85CFBEBB83245E3A">3.2-1 PermutMaxTries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80C15D427EC11A5F">3.2-2<
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8244918D7D1C2703">3.2-3 OneSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </spanstatic std:s getDescription;
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X826D28757824F83E">3.2-5 OneFSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BA6690983912C09">3.2-6 IsSPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83E103AF7C3DC9B7">3.2-7 OneSylowSubgroupNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86A62A52831BE00E">3.2-8 IsSNPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83D1BF5778D6790A">3.2-9 OneSystemNormaliserNotPermutingWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">  </        void testRunStarting(TestRunInfo const& runInfo) override
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7AC58CFF87770C6C">3.2-12 IsWeaklySPermutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href=<
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85E85FBF78DAB08E">3.2-14 IsWeaklyPermutable</a></span>
<span class="ContSS"><brvoid testGroupStarting(GroupInfoconst )override;
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84BE89EF8140D2C0">3.2-16 IsWeaklyNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X872EA72A7CEC3E65">3.2-18 IsWithSubnormalizerCondition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FBEC9DD7C8AED52">3.2-19 OneSubgroupInWhichSubnormalNotNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87CA542C7E1FEEF3">3.2-20 IsWithSubpermutizerCondition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79648B4F8758E187">3.2-21 OneSubgroupInWhichSubnormalNotPermutable</a></span>
<span         boo assertionEndedAssertionStats&java.lang.StringIndexOutOfBoundsException: Range [66, 64) out of bounds for length 75
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7AC9002B7D5AB2DA">3.2-23 OneSubgroupInWhichSubnormalNotSPermutable</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Permutability of Subgroups in Finite Groups</span></h3>

<p>This chapter describes functions to check permutability of subgroups in a given group. First we present a function to check whether a subgroup permutes with another one, then we present functions to test whether a java.lang.StringIndexOutOfBoundsException: Range [0, 225) out of bounds for length 75

<p><a id="X834D77E17960C0B7" name="X834D77E17960C0B7"></a></p>

<h4>3.1 <span class="Heading">Permutability functions</span></h4>

<p><a id="X862727A87C26B10B" name="java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 47

<h5>3.1-1 ArePermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>This function returns <code class="keyw">true</code> if <var class="Arg">U</var> and <var class="Arg">V</var> permute in <var class="Arg">G</var>. The groups <java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 74

<p>This is done by checking that the order of <span class="SimpleMath">⟨ U, V ⟩

<p>In the version with two arguments, <var class="Arg">U</var> and <var class="Arg">V</var> must have a common parent or <code class="code">ClosureGroup( U, V )</code> (see <code class="func">ClosureGroup</code> (


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ std:string  java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
<span class="GAPprompt">gap></span> <span class="GAPinputSectionNode  java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
Group([ (1,2)(3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=Subgroup(g,[(                           testOkToFail)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
Group([ (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Group([ (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">ArePermutableSubgroups(g,a,b);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ArePermutableSubgroups(g,a,c);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ArePermutableSubgroups(g,b,c);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ArePermutableSubgroups(b,c);</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">ArePermutableSubgroups(b,a);</span>
false
</pre></div>

<p><a id="X875BEB9B7D0DBC9A" name="X875BEB9B7D0DBC9A"></a></p>

<h4>3.2 <span class="Heading">Embedding properties related to permutability</span></h4>

<p>In the following we describe some functions which allow us to test whether a subgroup permutes with the members of some families of subgroups. We pay special attention toXmlWriter ;

<p><a id="X85CFBEBB83245E3A" name="X85CFBEBB83245E3A"></a></p>

<h5>3.2-1 PermutMaxTries</h5>

<div class="func"><table class="func" width="100%"><tr><        Timer suiteTimerjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<p>This variable contains the maximum number of random attempts of permutability checks before trying general java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 35

<p><a id="X80C15D427EC11A5F" name="X80C15D427EC11A5F"></a></p>

<h5>3.2-2 IsPermutable</h5>

< class=""<class""width="100"<<=t>< =">#;</code>( <var =A><var>,var=A">/java.lang.StringIndexOutOfBoundsException: Range [172, 171) out of bounds for length 246
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPermutableInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property returns <code class="keyw">true</code> if the subgroup <var class="Arg">H</var> is permutable in <var class="Arg">G</var>, otherwise it returns <code class="keyw">false</code>. We say that a subgroup <span class="SimpleMath">H</        bool  =;

<p>If the 

<p><a id="X8244918D7D1C2703" name="X8244918D7D1C2703"></a></p>

<h5>3.2-3 OneSubgroupNotPermutingWith</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227;
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSubgroupNotPermutingWithInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td>}//end namespaceCatch
<p>This attribute finds a cyclic subgroup of <var class="Arg">G</var> which does not permute


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ]/end java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 29
<span  h
Group([ (1,2)(3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=Subgroup(g,[(1,2,3)]);</span>
Group([ (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=Subgroup(g,[(1,2)]);</span>
Group([ (1,2) ])
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutable(g,b);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutable(g,c);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubgroupNotPermutingWith(g,b);</span>
Group([ (1,3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">v:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);</span>
Group([ (1,2)(3,4), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubgroupNotPermutingWith(g,v);</span>
fail
<span    :
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutable(g,v);</span>
true
</pre></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(16XmlReporterReporterConfig&_onfig;
<pc group of size 16 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Subgroup(g,[g.2]);</span>
Group([ f2 ])
<span class="GAPprompt">gap></span> <span class=
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutable(g,h);</span>
true
</pre></div>

<p>Sometimes one does not require a subgroup to permute with all subgroups of the group, but only with a selected family of subgroups of the group. The general case is the following.</p>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>3.2-4 IsFPermutable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsFPermutable</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>In this function, <var class="Arg">H</var> is a subgroup of <var class="Arg">G</var> and <var class="Arg">f</var> must be a list of subgroups of <var class="Arg">G</var>. It returns <code class="keyw">true</code> if <var class="Arg">H</var> permutes

<java.lang.StringIndexOutOfBoundsException: Range [16, 2) out of bounds for length 860

<p><a id="X826D28757824F83E" name="X826D28757824F83E"></a></p>

<h5>3.2-5 OneFSubgroupNotPermutingWith</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneFSubgroupNotPermutingWith</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>In this operation, <var class="Arg">H</var> is a subgroup of <var class="Arg">G</var> and <var class=    public:// StreamingReporterBase

<p>This function tries to use the values of <code class="func">IsPermutableInParent</code> (<a href="chap3.html#X80C15D427EC11A5F"><span class="RefLink">3.2-2</span></a>) and <code class="func">OneSubgroupNotPermutingWithInParent</code> (<a href="chap3.html#X8244918D7D1C2703"><span class="RefLink">3.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="
Group([ (1,2,3,4), (1,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(a);</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">OneFSubgroupNotPermutingWith(g,a,MaximalSubgroups(g));</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFPermutable(g,a,MaximalSubgroups(g));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">HasIsPermutableInParent(a);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutableInParent(a);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">HasOneSubgroupNotPermutingWithInParent(a);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubgroupNotPermutingWithInParent(a);</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFPermutable(g,a,AllSubnormalSubgroups(g));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">OneFSubgroupNotPermutingWith(g,a,AllSubnormalSubgroups(g));</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">sylows:=g->Union(List(SylowSubgroups(g),</span>
<span class="GAPprompt">></span> <span class="GAPinput">          t->ConjugacyClassSubgroups(g,t)));</span>
function( g ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">OneFSubgroupNotPermutingWith(g,a,sylows(g));</span>
Group((3,4) (,(2,),13(,)]java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
</pre></div>

<p>The following functions can be considered as particular cases of the previous function for some subgroup embedding functors. However, they can be

<p><a id="X7BA6690983912C09" name="X7BA6690983912C09"></a></p>

<h5>3.2-6 IsSPermutable</h5>

<div class="func"><table class="void sectionStarting(SectionInfo&) ;
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSPermutableInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This operation returns <java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(8,3);</span>
& 8 with3generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSPermutable(g,Subgroup(g,[g.1]));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutable(g,Subgroup(g,[g.1]));</span>
false
</pre></div>

<p><a id="X83E103AF7C3DC9B7" name="X83E103AF7C3DC9B7"></a></p>

<h5>3.2-7 OneSylowSubgroupNotPermutingWith</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSylowSubgroupNotPermutingWith</code>( <var class="Arg">G</var>, <var class="Arg">H</java.lang.StringIndexOutOfBoundsException: Range [0, 191) out of bounds for length 75
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSylowSubgroupNotPermutingWithInParent</code>( <var class="Argjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The argument <var class="Arg">H</var> must be a subgroup of rride;


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Subgroup(g,[(1,2)(3,4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSylowSubgroupNotPermutingWith(g,a);</span>
Group([ (2,4,3) ])
</pre></div>

<p><a id="X86A62A52831BE00E" name="X86A62A52831BE00E"></a></p>

<h5>3.2-8 IsSNPermutable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSNPermutable</code>( <var class="Arg">G</var>, <var class="Arg">H</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">&#an style='color: green'>8227; IsSNPermutableInParent</code>( <var class="Arg">H</var> )        java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 75
<p>This operation returns <code class="keyw">true</code>         void testRunEnded(TestRunStats const&&nbsp;testRunStats) testRunEndedTestRunStatsconst& testRunStats) override;

<p><a id="X83D1BF5778D6790A" name="X83D1BF5778D6790A"></a></p>

<h5>3.2-9 OneSystemNormaliserNotPermutingWith</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSystemNormaliserNotPermutingWith</code>( <var class="Arg">G</var>, <var class="Arg">H</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">&#an style='color: green'>8227; OneSystemNormaliserNotPermutingWithInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute        void benchmarkPreparing(td:string const&name override;
<p>Here <var class="Arg">G</var> must be a soluble group and <var class="Arg">H</var> must be a subgroup of <var class="Arg">G</var>. If <var class="Arg">H</var> permutes with all system normalisers of <var class="Arg">G</var>, then this operation returns <code class="keyw">fail</code>. Otherwise, it returns a         void benchmarkStarting(BenchmarkInfo const&) override;


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(4,5,6),(1,2));</span>
Group([ (1,2,3), (4,5,6), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Subgroup(g,[(1,2,3)(4,5,6)]);</span>
(1,)4,6 )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSNPermutable(g,a);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSPermutable(g,a);</span>
false
</pre></div>

<p>The next functions are not particular cases of <code class="func">IsFPermutable</code> (<a href="chap3.html#X865C7F81823FFA10"><span class="RefLink">3.2-4</span></a>) or <code class="func">OneFSubgroupNotPermutingWith</code> (<a java.lang.StringIndexOutOfBoundsException: Index 234 out of bounds for length 12

<p><a id="X7B2E052E7F9C4251" name="X7B2E052E7F9C4251"></a></p>

<h5>3.2-10 IsConjugatePermutable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsConjugatePermutable</code>( <var class="Arg">G</var>, <var class="Arg">H</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">&#an style='color: green'>8227; IsConjugatePermutableInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0


<div class="example"><pre}// end
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Subgroup(g,[(1,2)(3,4)]);</span>
([(1,2)(,)])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutable(g,a);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</pre></div>

<p><a id=" catch_external_interfaces

<h5>3.2-11 OneConjugateSubgroupNotPermutingWith</h5>

<div class="func"><table class="func" #ndif
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneConjugateSubgroupNotPermutingWithInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This operation finds a conjugate#f defined(CATCH_CONFIG_ENABLE_BENCHMARKING)


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(16,7);</span>
<pc group of size 16 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Subgroup(g,[g.1*g.4]);</span>
Group(
<span class="GAPprompt">gap></span> <span class="GAPinput">IsConjugatePermutable(g,h);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OneConjugateSubgroupNotPermutingWith(g,h);</span>
Group([f1*]java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
</pre></div>

<p>Next we introduce some subgroup embedding conciseinclude of thebenchmarkingfeatures shouldpreferthe

<p><a id="X7AC58CFF87770C6C" name="X7AC58CFF87770C6C"></a></p>

<h5>3.2-12 IsWeaklySPermutable</h5>

<div class="func"><table  individual includes in standard.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWeaklySPermutableInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property )</td></tr></table></div>
p>   bythis  is < class="></ode>when var class="Arg></> is a<>weakly-permutable/m subgroup of varclass"rg"G/ar,thatis spanclass""H/span Spermutablein< class="SimpleMath">⟨ H, H^g ⟩</span> implies that <span class="SimpleMath">H</span> is S-permutable in <span class="SimpleMath">⟨ H, g ⟩</span>, and <code class="keyw">false</code> otherwise.</p>

<p><a id="X870923B4798E5097" name="X870923B4798E5097"></a></p>

<h5>3.2-13 OneElementShowingNotWeaklySPermutable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneElementShowingNotWeaklySPermutable</code>( <var class="Arg">G</var
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneElementShowingNotWeaklySPermutableInParent</code>( <var class="Arg">/ start catch_chronometer.hpp
<p>If <var class="Arg">H</var> is a weakly S-permutable subgroup of <var class="Arg">G</var>, then this operation returns <code class="keyw">fail</code>. Otherwise, the value returned by this operation is an element <span class="SimpleMath">g ∈ G</span> such that <var class="Arg">H</var> is S-permutable in <span class="SimpleMath">⟨ H, H^g ⟩</span>, but <span class="

<p><a id="X85E85FBF78DAB08E" name="X85E85FBF78DAB08E"></a></p>

<h5>3.2-14 IsWeaklyPermutable

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWeaklyPermutable</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This operation returns/ Clocks

<p><a id="X8427C8DD78732D56" name="X8427C8DD78732D56"></a></p>

<h5>3.2-15 OneElementShowingNotWeaklyPermutable</h5>

<div class="func"><table class="func
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneElementShowingNotWeaklyPermutableInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">H</var> is a weakly permutable java.lang.StringIndexOutOfBoundsException: Range [0, 62) out of bounds for length 17

<p><a id="X84BE89EF8140D2C0" name="X84BE89EF8140D2C0"></a></p>

<h5>3.2-16 IsWeaklyNormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWeaklyNormal</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td   {
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWeaklyNormalInParent</code>( <var class="Arg"    namespace Benchmark {
<p>This operation returns <code class="keyw">true</code> if <var class="Arg">H</var> is weakly normal in <var class="Arg">G</var>, and <code class="keyw">false</code> otherwise. A subgroup <span class="SimpleMath">H</span> of <span java.lang.StringIndexOutOfBoundsException: Index 234 out of bounds for length 33

<p><a id="X82C8AE027E578ACF" name="X82C8AE027E578ACF"></a></p>

<h5>3.2-17 OneElementShowingNotWeaklyNormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneElementShowingNotWeaklyNormal</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Range [0, 85) out of bounds for length 0
<p>If <var class="Arg">H</var> is a weakly normal subgroup of <var class="Arg">G</var>, then this function        template <typename Clock>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=DihedralGroup(8);</span>
<pc group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Subgroup(g,[g.1]);</span>
Group([ f1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWeaklySPermutable(g,a);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWeaklyPermutable(g,a);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=        using default_clock = std::chrono::steady_clock;
f2
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(Normalizer(g,a),ConjugateSubgroup(a,x));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x in Normalizer(g,a);</span>
false
</pre></div>

<p><a id="X872EA72A7CEC3E65" name="X872EA72A7CEC3E65"></ template  <typenameClockjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

<h5>3.2-18 IsWithSubnormalizerCondition</h5>

<div class="unc><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWithSubnormalizerCondition</code>( <var class="Arg">G</var>, <var class="Arg">H</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">&#an style='color: green'>8227; IsWithSubnormalizerConditionInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227;                  Clock:)
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWithSubnormaliserConditionInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This operation returns <code class="keyw">true</code> if the        }

<p>A subgroup <span class="SimpleMath">H</span> is said to <em>satisfy the subnormaliser condition</em> in <span class="SimpleMath">G</span> if the condition that <span class="SimpleMath">H</span> is subnormal in a subgroup <span class="SimpleMath">K</span> of <span class="SimpleMath">G</span> implies that <span class="SimpleMath">H</span> is normal in <span class="SimpleMath">K</span>.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<=" =">/>/>

<h5>3.2-19 OneSubgroupInWhichSubnormalNotNormal</h5>

<div}//  Catch
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSubgroupInWhichSubnormalNotNormalInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function returns a subgroup <span class="SimpleMath">K</span> of <span class="SimpleMath">G</span> such <p>This function returns a subgroup <span class="SimpleMath">K</span> of <span class="SimpleMath">G</span> such that

<p><a id="X87CA542C7E1FEEF3" name="X87CA542C7E1FEEF3"></a></p>

<java.lang.StringIndexOutOfBoundsException: Range [28, 3) out of bounds for length 44

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWithSubpermutizerCondition</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td><
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWithSubpermutizerConditionInParent
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWithSubpermutiserCondition</code>( <var class="Arg">G</var>, <var class="Arg">H</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">&#an style='color: green'>8227; IsWithSubpermutiserConditionInParent()
<p>This operation returns <code class="keyw">true</code> if the subgroup <span class="SimpleMath">H</span> satisfies the subpermutiser condition in <span class="SimpleMath">G</span>, and <code class="keyw">false</code> otherwise.</p>

<>A spanclass=SimpleMath></pan>issaidtoemsatisfyjava.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 397

<p><a id="X79648B4F8758E187" name="X79648B4F8758E187"></a></p>

<h5>3.2-21 OneSubgroupInWhichSubnormalNotPermutable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSubgroupInWhichSubnormalNotPermutable</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</java.lang.StringIndexOutOfBoundsException: Range [0, 254) out of bounds for length 6
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="namespace  {
<p>This function returns a subgroup <span class="SimpleMath">K</span> of <span class="SimpleMath">G</span> such    namespace Benchmark

<p><a id="X831F5B8787BB111F" name="X831F5B8787BB111F"></a></p>

<h5>3.<fdefined(_NUC__ |defined(__lang__

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWithSSubpermutizerCondition</code>( <var class="Arg">G</var>, <var class="Arg">H</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">&#an style='color: green'>8227; IsWithSSubpermutizerConditionInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( property&<       inline  keep_memoryT*) java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsWithSSubpermutiserCondition</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%inline void keep_memory() {
<p>This operation returns <code class="keyw">true</code> if the subgroup <span class="SimpleMath">H</span> satisfies the S-subpermutiser condition in <span class="SimpleMath">G</span>, and <code class="keyw">false</code> otherwise.</p>

<p>A subgroup <span class="SimpleMath">H</span> is said to <em>satisfy the S-subpermutiser condition</em> in <span class="SimpleMath">G</span> if the condition that <span class="SimpleMath">H</span> is subnormal in a subgroup <span class="SimpleMath">K</span> of <span class="SimpleMath">G</span> implies that <span class="SimpleMath">H</span> is S-permutable in <span class="SimpleMath">K</span>.</p>

<><aid"" name"X7AC9002B7D5AB2DA"<a/

<h5>3.2-23 OneSubgroupInWhichSubnormalNotSPermutable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code<           inline void optimizer_barrier( {keep_memory)}
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OneSubgroupInWhichSubnormalNotSPermutableInParent</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function returns a subgroup <span class="SimpleMath">K</span> of <span class="SimpleMath">G</span> such that <span class="SimpleMath">H</span> is subnormal in <span class="SimpleMath">K</span> and <span class="SimpleMath">H</span> is not S-permutable in <span class="SimpleMath">K</span> if #lifdefined()


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(324,160);</span>
<pc group of size 324 with 6 generators>
<span class="GAPprompt">gap></span> <span#ragmaoptimize(",off)
Group([ f3, f5 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWithSubnormalizerCondition(g,a);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWeaklyNormal(g,a);</span>
false
<spanclass"APprompt>java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 94
false
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=OneElementShowingNotWeaklyNormal(g,a);</span>
f1
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugateSubgroup(a,x)=a;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(Normalizer(g            *einterpret_cast<char volatile*>(p) = *reinterpret_cast<char const volatile*>(p);
true
</pre></div>


<div class="chlinkprevnextbot"> <a         }


<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>  <        / TODO equivalent keep_memory()

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck            inline void optimizer_barrier() {
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