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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap74.html">[Previous Chapter]</a>    <a href="chap76.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap75_mj.html">[MathJax on]</a></p>
<p><a id="X80D9CA647E680B19" name="X80D9CA647E680B19"></a></p>
<div class="ChapSects"><a href="chap75.html#X80D9CA647E680B19">75 <span class="Heading">Monomiality Questions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap75.html#X7F2F753F7B354F09">75.1 <span class="Heading">InfoMonomial (Info Class)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X8103DD607C7F2CD2">75.1-1 InfoMonomial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap75.html#X842D10BC7CA9C2DE">75.2 <span class="Heading">Character Degrees and Derived Length</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X86A900897819E5AC">75.2-1 Alpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X82C33CF282FC5A73">75.2-2 Delta</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X7D0D26267A9D37DD">75.2-3 <span class="Heading">IsBergerCondition</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap75.html#X82C21037806B52CE">75.3 <span class="Heading">Primitivity of Characters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X81FD26947924C500">75.3-1 TestHomogeneous</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X7BC72ECE822D4245">75.3-2 IsPrimitiveCharacter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X82BFA6968415F308">75.3-3 TestQuasiPrimitive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X84860E3A7FECDBA3">75.3-4 TestInducedFromNormalSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap75.html#X86567D7F781BBCAE">75.4 <span class="Heading">Testing Monomiality</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X84EB92B57DAF5C93">75.4-1 <span class="Heading">TestMonomial</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X787CCCBB7FC17F5E">75.4-2 TestMonomialUseLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X8261B5AA7BCFFCC2">75.4-3 IsMonomialNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X822E03EF7B8F92D3">75.4-4 <span class="Heading">TestMonomialQuick</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X7E56A0EA868CC34A">75.4-5 <span class="Heading">TestSubnormallyMonomial</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X83EF7B8D7C1C2CA3">75.4-6 <span class="Heading">TestRelativelySM</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap75.html#X7B5735897DE29BCB">75.5 <span class="Heading">Minimal Nonmonomial Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X7D7E2667821A23CD">75.5-1 IsMinimalNonmonomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap75.html#X7B416BBD80072079">75.5-2 MinimalNonmonomialGroup</a></span>
</div></div>
</div>

<h3>75 <span class="Heading">Monomiality Questions</span></h3>

<p>This chapter describes functions dealing with the monomiality of finite (solvable) groups and their characters.</p>

<p>All these functions assume <em>characters</em> to be class function objects as described in Chapter <a href="chap72.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>, lists of character <em>values</em> are not allowed.</p>

<p>The usual <em>property tests</em> of <strong class="pkg">GAP</strong> that return either <code class="keyw">true</code> or <code class="keyw">false</code> are not sufficient for us. When we ask whether a group character <span class="SimpleMath">χ</span> has a certain property, such as quasiprimitivity, we usually want more information than just yes or no. Often we are interested in the reason <em>why</em> a group character <span class="SimpleMath">χ</span> was proved to have a certain property, e.g., whether monomiality of <span class="SimpleMath">χ</span> was proved by the observation that the underlying group is nilpotent, or whether it was necessary to construct a linear character of a subgroup from which <span class="SimpleMath">χ</span> can be induced. In the latter case we also may be interested in this linear character. Therefore we need test functions that return a record containing such useful information. For example, the record returned by the function <code class="func">TestQuasiPrimitive</code> (<a href="chap75.html#X82BFA6968415F308"><span class="RefLink">75.3-3</span></a>) contains the component <code class="code">isQuasiPrimitive</code> (which is the known boolean property flag), and additionally the component <code class="code">comment</code>, a string telling the reason for the value of the <code class="code">isQuasiPrimitive</code> component, and in the case that the argument <span class="SimpleMath">χ</span> was <em>not</em> quasiprimitive also the component <code class="code">character</code>, which is an irreducible constituent of a nonhomogeneous restriction of <span class="SimpleMath">χ</span> to a normal subgroup. Besides these test functions there are also the known properties, e.g., the property <code class="func">IsQuasiPrimitive</code> (<a href="chap75.html#X82BFA6968415F308"><span class="RefLink">75.3-3</span></a>) which will call the attribute <code class="func">TestQuasiPrimitive</code> (<a href="chap75.html#X82BFA6968415F308"><span class="RefLink">75.3-3</span></a>), and return the value of the <code class="code">isQuasiPrimitive</code> component of the result.</p>

<p>A few words about how to use the monomiality functions seem to be necessary. Monomiality questions usually involve computations in many subgroups and factor groups of a given group, and for these groups often expensive calculations such as that of the character table are necessary. So one should be careful not to construct the same group over and over again, instead the same group object should be reused, such that its character table need to be computed only once. For example, suppose you want to restrict a character to a normal subgroup <span class="SimpleMath">N</span> that was constructed as a normal closure of some group elements, and suppose that you have already computed with normal subgroups (by calls to <code class="func">NormalSubgroups</code> (<a href="chap39.html#X80237A847E24E6CF"><span class="RefLink">39.19-9</span></a>) or <code class="func">MaximalNormalSubgroups</code> (<a href="chap39.html#X82ECAA427C987318"><span class="RefLink">39.19-10</span></a>)) and their character tables. Then you should look in the lists of known normal subgroups whether <span class="SimpleMath">N</span> is contained, and if so you can use the known character table. A mechanism that supports this for normal subgroups is described in <a href="chap71.html#X81272CEE79F13E7B"><span class="RefLink">71.23</span></a>.</p>

<p>Also the following hint may be useful in this context. If you know that sooner or later you will compute the character table of a group <span class="SimpleMath">G</span> then it may be advisable to compute it as soon as possible. For example, if you need the normal subgroups of <span class="SimpleMath">G</span> then they can be computed more efficiently if the character table of <span class="SimpleMath">G</span> is known, and they can be 

<thisjava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 262


j:esetf)/java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<span class="GAPprompt">gap></span> <span class="input type="radio" name="colorprompt" value="" checked="checked
<java.lang.StringIndexOutOfBoundsException: Range [50, 5) out of bounds for length 88
</pre></div>

<pDisplay side table ofcontents within chapters: 

<h4>75.1 <span class="Heading">InfoMonomial (Info Class)</span></h4>

<p><a id="type="radio" name="font" value="" checked="checked"

<h5>75. 

<div class="func"><table class="func" width="100:f)"/ right 
< functions in  chapterprintsome hopefully useful) <em>information</em> if the info level of the info class <code class="func">InfoMonomial</code> is at least <span class="SimpleMath">1/p

<p><a id="X842D10BC7CA9C2DE" name="X842D10BC7CA9C2DE"></a></p>

<h4>75.2 <span class="Heading">Character Degrees and Derived Length</span></h4>

<p><a id="X86A900897819E5AC" name="X86A900897819E5AC"></a></p>

<h5>75.2-1 Alpha</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Alpha</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a group <var class="Arg">G</var>, <code class="func">Alpha</code> returns a list whose <span class="SimpleMath">i</span>-th entry is the maximal derived length of groups <span class="SimpleMath"><var class="Arg">G</var> / ker(χ)</span> for <span class="SimpleMath">χ ∈ Irr(<var class="Arg">G</var>)</span> with <span class="SimpleMath">χ(1)</span> at most the <span class="SimpleMath">i</span>-th irreducible degree of <var class="Arg">G</var>.</p>

<p><a id="X82C33CF282FC5A73" name="X82C33CF282FC5A73"></a></p>

<h5>75.2-2 Delta</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Delta</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a group <var class="Arg">G</var>, <code class="func">Delta</code> returns the list <span class="SimpleMath">[ 1, alp[2] - alp[1], ..., alp[<var class="Arg">n</var>] - alp[<var class="Arg">n</var>-1] ]</span>, where <span class="SimpleMath">alp =</span><code class="code">Alpha( <var class="Arg">G</var> )</code> (see <code class="func">Alpha</code> (<a href="chap75.html#X86A900897819E5AC"><span class="RefLink">75.2-1</span></a>)).</p>

<p><a id="X7D0D26267A9D37DD" name="X7D0D26267A9D37DD"></a></p>

<h5>75.2-3 <span class="Heading">IsBergerCondition</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsBergerCondition</code>( <var class="Arg">G</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; IsBergerCondition</code>( <var class="Arg">chi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Called with an irreducible character <var class="Arg">chi</var> of a group <span class="SimpleMath">G</span>, <code class="func">IsBergerCondition</code> returns <code class="keyw">true</code> if <var class="Arg">chi</var> satisfies <span class="SimpleMath">M' ≤ ker(χ)</span> for every normal subgroup <span class="SimpleMath">M</span> of <span class="SimpleMath">G</span> with the property that <span class="SimpleMath">M ≤ ker(ψ)</span> holds for all <span class="SimpleMath">ψ ∈ Irr(G)</span> with <span class="SimpleMath">ψ(1) < χ(1)</span>, and <code class="keyw">false</code> otherwise.</p>

<p>Called with a group <var class="Arg">G</var>, <code class="func">IsBergerCondition</code> returns <code class="keyw">true</code> if all irreducible characters of <var class="Arg">G</var> satisfy the inequality above, and <code class="keyw">false</code> otherwise.</p>

<p>For groups of odd order the result is always <code class="keyw">true</code> by a theorem of T. R. Berger (see <a href="chapBib.html#biBBer76">[Ber76, Thm. 2.2]</a>).</p>

<p>In the case that <code class="keyw">false</code> is returned, <code class="func">InfoMonomial</code> (<a href="chap75.html#X8103DD607C7F2CD2"><span class="RefLink">75.1-1</span></a>) tells about a degree for which the inequality is violated.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Alpha( Sl23 );</span>
[ 1, 3, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Alpha( S4 );</span>
[ 1, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Delta( Sl23 );</span>
[ 1, 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Delta( S4 );</span>
[ 1, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBergerCondition( S4 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBergerCondition( Sl23 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">List( Irr( Sl23 ), IsBergerCondition );</span>
[ true, true, true, false, false, false, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( Irr( Sl23 ), Degree );</span>
[ 1, 1, 1, 2, 2, 2, 3 ]
</pre></div>

<p><a id="X82C21037806B52CE" name="X82C21037806B52CE"></a></p>

<h4>75.3 <span class="Heading">Primitivity of Characters</span></h4>

<p><a id="X81FD26947924C500" name="X81FD26947924C500"></a></p>

<h5>75.3-1 TestHomogeneous</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestHomogeneous</code>( <var class="Arg">chi</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a group character <var class="Arg">chi</var> of a group <span class="SimpleMath">G</span> and a normal subgroup <var class="Arg">N</var> of <span class="SimpleMath">G</span>, <code class="func">TestHomogeneous</code> returns a record with information whether the restriction of <var class="Arg">chi</var> to <var class="Arg">N</var> is homogeneous, i.e., is a multiple of an irreducible character.</p>

<p><var class="Arg">N</var> may be given also as list of conjugacy class positions w.r.t. the character table of <span class="SimpleMath">G</span>.</p>

<p>The components of the result are</p>


<dl>
<dt><strong class="Mark"><code class="code">isHomogeneous</code></strong></dt>
<dd><p><code class="keyw">true</code> or <code class="keyw">false</code>,</p>

</dd>
<dt><strong class="Mark"><code class="code">comment</code></strong></dt>
<dd><p>a string telling a reason for the value of the <code class="code">isHomogeneous</code> component,</p>

</dd>
<dt><strong class="Mark"><code class="code">character</code></strong></dt>
<dd><p>irreducible constituent of the restriction, only bound if the restriction had to be checked,</p>

</dd>
<dt><strong class="Mark"><code class="code">multiplicity</code></strong></dt>
<dd><p>multiplicity of the <code class="code">character</code> component in the restriction of <var class="Arg">chi</var>.</p>

</dd>
</dl>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= DerivedSubgroup( Sl23 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">chi:= Irr( Sl23 )[7];</span>
Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">test:= TestHomogeneous( chi, n );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">test.isHomogeneous;  test.comment;  test.multiplicity;</span>
false
"restriction checked"
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Degree( test.character );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">chi:= Irr( Sl23 )[4];</span>
Character( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">cln:= ClassPositionsOfNormalSubgroup( CharacterTable( Sl23 ), n );</span>
[ 1, 4, 7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TestHomogeneous( chi, cln );</span>
rec( comment := "restricts irreducibly", isHomogeneous := true )
</pre></div>

<p><a id="X7BC72ECE822D4245" name="X7BC72ECE822D4245"></a></p>

<h5>75.3-2 IsPrimitiveCharacter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPrimitiveCharacter</code>( <var class="Arg">chi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>For a character <var class="Arg">chi</var> of a group <span class="SimpleMath">G</span>, <code class="func">IsPrimitiveCharacter</code> returns <code class="keyw">true</code> if <var class="Arg">chi</var> is not induced from any proper subgroup, and <code class="keyw">false</code> otherwise. This currently only works for characters of soluble groups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimitiveCharacter( Irr( Sl23 )[4] );</span>
true2 java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 22
<< classfunc>tableclass"""100"trtd class="tdleft"><code class="func">&#8227; GeneratorsOfPresentation</code>( <var class="Arg><var> <td><class="dright>(nbsp;n;<td>/tr>/>/div>
false
</pre></div>

<p><a id="X82BFA6968415F308" name="X82BFA6968415F308"></a></p>

<h5>75.3-3 TestQuasiPrimitive</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestQuasiPrimitive</code>( <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< func"><class=funcwidth"00%>tr>td =""codeclassfunc"&8227; <code <class""chi<var)/td>tdclass="tdright">( property )</td></tr></table></div>
<p><code class="func">TestQuasiPrimitive</code> returns a record with information about quasiprimitivity of the group character <var class="Arg">chi</var>, i.e., whether <var class="Arg">chi</var> restricts homogeneously to every normal subgroup of its group. The result record contains at least the components <code class="code">isQuasiPrimitive</code> (with <>java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 102

<>codeclass""IsQuasiPrimitive</code>returns code class=keyw></  codeclass""<code>  on whether the character <var class="Arg">chi</var> is quasiprimitive.</p>

<p>Note that for java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 5


<div class="example"><pre>
;</span> < java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 90
Character(div ""< %>tr >codeclassjava.lang.StringIndexOutOfBoundsException: Range [93, 91) out of bounds for length 290
<span classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 95
rec( comment := "all restrictions checked", isQuasiPrimitive := true )
<span class="GAPprompt">gap></span> <span class="GAPinput">chi:= Irr( Sl23 )[7];</span>
Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">test:= TestQuasiPrimitive( chi );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">test.isQuasiPrimitive;  test.comment;</span>
false
"restriction checked"
<span class=G>gap><span> span ""egreetest.character );</span>
1
</pre></div>

<p><a id="X84860E3A7FECDBA3"<>operation<class"func>java.lang.StringIndexOutOfBoundsException: Range [123, 54) out of bounds for length 385

<h5>75.3-4 TestInducedFromNormalSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestInducedFromNormalSubgroup</code>( <var class="Arg">chi</var>[, <var class="Arg">N</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsInducedFromNormalSubgroup</code>( <var class="Arg">chi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">TestInducedFromNormalSubgroup</code> returns a record with information whether the irreducible java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The result contains always the components <code class="code">isInduced</code> (either< ="gap&t&lt;/pan><span class"GAPinput" :=GeneralLinearGroup( 2  )<span>

<>code class=func"IsInducedFromNormalSubgroup</code> returns <code class="keyw">true</code> if <var class="Arg">chi</var> is induced from a proper normal subgroup of <span class="SimpleMath">G</span>, and <code class="keyw">false</code> otherwise.</p>


<=e"<>
<span2016
[ false, false ,false,false,false,true]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( Irr( S4 ){ [ 1, 3, 4 ] },</span>
<pan class="GAPprompt>gt;/span < =GAPinput"         TestInducedFromNormalSubgroup;</>
[ rec( comment := "linear character", isInduced := false ),
  rec( character := Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
        [ 1, 1, E(3), E(3)^2 ] ),
      comment := "induced from component '.character'",
      isInduced := true ),
  rec( comment := "all maximal normal subgroups checked",
      isInduced := false ) ]
</pre></div>

<p><a id="X86567D7F781BBCAE" name="X86567D7F781BBCAE"></a></p>

<h4>4<class="eading>Testing Monomiality</span><h4>

<p>A character <span java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 0

<p><a id="X84EB92B57DAF5C93

-<java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 57

<div class="func"><table class="unc"width="100%"><tr><td class="tdleft> java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 222
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestMonomial</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><<free group on the gea b c &;
<><tr><td "dleft>code class="func">‣ TestMonomial</code>( <var class="Arg">G</var>, <var class="Arg">uselattice</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Called with a group character <var class="Arg">chi/var>of group<varclass=Arg">G<var <code class="unc">TestMonomial</code> returns a information  of the group <var class="Arg">G</var> or the group character <var class="Arg">chi</var>, respectively.</p>

<p>If <code class="func"> java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 99

<If <code class="func">TestMonomial</code> proves <var class="Arg">chi</var> or <var class="Arg">G</var> to be nonmonomial then the value of the component <code class="code">isMonomial</code> is <code class="keyw">false</code>, and in the case of <var class="Arg">G</var> a nonmonomial character is the value of the component <code class="code">character</code> if it had been necessary to compute it.</p>

<p>A Boolean can a^2b*2*-*1**(1*^-1^,

<p>For a group whose java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 731


<div class="example"><java.lang.StringIndexOutOfBoundsException: Range [0, 25) out of bounds for length 2
<span class="GAPprompt">gap></span> <span <>it  java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 886
rec( comment := "abelian by supersolvable group", isMonomial := true)
<span class="GAPprompt">gap></span> <span class="
rec comment =listDeltaG containsentry 1java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
  isMonomial := false )
</pre></div>

<p><a id="X787CCCBB7FC17F5E" name="X787CCCBB7FC17F5E"></a></p>

<h5>75.4-2 TestMonomialUseLattice</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestMonomialUseLattice</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This global variable controls for which groups the operation <code class="func"><span class="GAPprompt">gap></span> <span class="GAPinput">P := PresentationViaCosetTable>

<p><a id="X8261B5AA7BCFFCC2" name="X8261B5AA7BCFFCC2"></a></p>

<h5>75.4-3 IsMonomialNumber</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsMonomialNumber</code>( <var class="Arg">n</var> )</td><td class="tdright">( property )</td></tr></table></div>
< java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 318

<p>Let <span class="SimpleMath">ν_p(n)</span> denote the multiplicity of the prime <span class="SimpleMath">p</span> as factor of <span class="SimpleMath">n</span>, and <span class="SimpleMath">ord(p,q)</span> the multiplicative order of <span class="SimpleMath">p mod q</span>.</p>

<p>Then there exists a solvable p><a id="X7E1F2658827FC228" name="X7E1F2658827FC228


<dl>divclass"">tableclass=func"width="100%"><tr><td class="tdleft"><code class="func">‣ SimplifiedFpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<dt><strong class="Mark">1.</strong></dt>
<dd><p><span class="SimpleMath">ν_2(n) ≥ 2</span> and there is a <span class="SimpleMath">p</span> such that <span class="SimpleMath">ν_p(n) ≥ 3</span> and <span class="SimpleMath">p ≡ -1 mod 4</span>,</p>

</dd>
<dt><strong class="Mark">2.</strong>sclass=>gap;</span> <span class="GAPinput">TzPrint( P );</span>
<dd><p><span classjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 13

</dd>
<t< =java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 41
<dd><p>there are odd prime divisors <span class="SimpleMath">p</span> and <span class="SimpleMath">q</span> of <span class="SimpleMath">n</span> such that <span class="SimpleMath">ord(p,q)</#  68[ ,-2, 23 ,3 -,3 

</dd>
<t<trong "Mark"./trong>/t
<dd><p>there is a prime divisor <span class="

</dd>
<dt><strong class="Mark">5.</strong></div java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 247
<dd><p><span class="SimpleMath">ν_2(n) ≥ 2</span> and there is a <span class="SimpleMath">p</span> such that <span class="SimpleMath">p ≡ 1 mod 4</span>, <p>prints the <var class="Arg">n</var> most often occurring relator subwords of the form <span class="SimpleMath">a b</span>, where <span class="SimpleMath">a<java.lang.StringIndexOutOfBoundsException: Range [218, 217) out of bounds for length 532

</dd>
</dl>
<p>These five possibilities correspond to the five pThefunction <class"unc"is in contextjava.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 363


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered( [ 1 .. 111 ], x ->java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 24, 48, 72, 96, 108 ]
<pre>/divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p><a id="X822E03EF7B8F92D3" name="X822E03EF7B8F92D3"></a></p>

<h5>75.4-4 <span class="Heading">TestMonomialQuick</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestMonomialQuick</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width<pre>d>
<p><code class="func">java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0


<dl>
<rong class=M>cclass="code">isMonomial</code></strong></dt>
<dd><p>either <code class="keyw">true</code> or <code class="keyw">false</code> or the string <code class="code">"?"</code>, depending on whether (non)monomiality could be proved, and</p>

</dd>
<dt><strong class="Mark"><code class="code">comment</code></strong></dt>
<dd><p>a string telling the reason for the value of the <code class="code">isMonomial</code> component.</p>

</dd>
</dl>
<p>A group <var class="Arg">G</var> is proved to be monomial by <code class="func">TestMonomialQuick</code> if <var class="Arg">Gjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>An irreducible character <var class="Arg">chi</var> is proved to be monomial if it is linear, or if its codegree is a prime power, or if its group knows to be monomial, or if the factor group modulo the java.lang.StringIndexOutOfBoundsException: Index 208 out of bounds for length 76


<div class="example"><pre>
<span class="GAPprompt">gap><
rec( comment := "whole group is monomialp>may  the java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 435
<=G">/span><pan =GAPinput">TestMonomialQuick( S4 ;/pan
rec( comment := "abelian by supersolvable group", isMonomial := true )
<span class="GAPprompt">gap></span> <span class="GAPinput">TestMonomialQuick( Sl23 );</span>
rec( comment := "no decision by cheap tests", isMonomial := "?" )
</pre></div>

<>a id=X7E56A0EA868CC34A"name="></<p>

<h5>75.4-5 <span class="Heading">TestSubnormallyMonomial</spanspan class"APprompt">&<pan> <class=GAPinput"H :=Subgroup(java.lang.StringIndexOutOfBoundsException: Range [80, 78) out of bounds for length 107

< class"func"<class=""width=100">tr><class=tdleft">cclass=f"&8227;; </code> < class="Arg">G</var>)/tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestSubnormallyMonomial</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attributeltpresentation 4gens and 7rels of total 21&;
<div class="func"><table class="func" width="100%"><tr><td class=I    has generators,the generatoris_7
<div class="func"><table class="func" width="100%"><tr><tdt>gens: (P)<span>
<p>An irreducible character of the group <span class="SimpleMath">G</spanjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 27

<p><code class="func">TestSubnormallyMonomial</code> returns a x7

<p>The result has the components <code class="code">isSubnormallyMonomial</code> (either <code class="keyw">true</java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 4

<p><code class="func">IsSubnormallyMonomial</code> returns <code class="keywp>a "83name"java.lang.StringIndexOutOfBoundsException: Range [55, 52) out of bounds for length 62


<div class="example">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pan ">gt;<span><pan =GAPinput>TestSubnormallyMonomial  ;/
rec( character := Character( CharacterTable( S4 ), [ 3, -1, -1 and it to the  <class=">/>  is calledby <ode class=func"></ <href="html#><pan class="RefLink">48.5-<span>a)and several Tietze transformation commands. As it does not know which global lists have to be kept consistent, you should not call it. Instead, you should call the function <code class="func">AddGenerator</code> (<a href="chap48.html#X7F632A6D8685855D"><span class="RefLink">48.5-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      ,comment ="ound non-character"
  isSubnormallyMonomial := false )
<span class="GAPprompt">gap></span> <span class="GAPinput">TestSubnormallyMonomial( Irr( S4 )[4] );</span>
rec( omment: "all  checked,
  isSubnormallyMonomial := false )
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 0
rec( comment := "all irreducibles checked",
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
<p<div>

<p><a id="X83EF7B8D7C1C2CA3premoves the<java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 167

<h5>75.4-6 <span class="Heading">TestRelativelySM</span></h5>

<div class=func><table class=func" "100%"<tr< class"tdleft>code class=func>#8227 TestRelativelySM<java.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 224
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TestRelativelySM</code>( <var class="Arg">chi</var> )</td><td class
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code 
< ">table class="unc" width"100 class=t>c class=func"&8227;<code> vclass="Arg"chi<v> var "rg>/ )td><td class="tdright">&bsp;&)/>/</<div>
<div class="func"><table class="func" width="100
<le ""100>tr>< "tdleft>cjava.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 223
<p>In the first two cases, <code class="func">TestRelativelySM</code> returns a record with information whether the argument, which must bejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>In the second two cases, a normal subgroup <var class="Arg">N</var> of <var class="java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 0

<p>The result record has the components <code class

<p><code class="func">IsRelativelySM</code> returns 

<p><em>Note<


<div =<java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class=pIfvar "Arg>java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 367
true
<span class="GAPprompt">gap></span> <span class="GAPinput">TestRelativelySM( DerivedSubgroup( S4 ) );</span>
rec(
  comment := "normal subgroups are abelian or have nilpotent factor gr\
:  )
</pre></div>

<p><a idfunc><java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 227

<h4>75.5 <span class="Heading">Minimal Nonmonomial Groups</span></h4>

<p><a id="X7D7E2667821A23CD" name="X7D7E2667821A23CD"></a></p>

<h5>75.5-1 IsMinimalNonmonomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsMinimalNonmonomial</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group <var class="Arg">G</var> is called <em>minimal nonmonomial</em> if it is nonmonomial, and all proper subgroups and factor groups are monomial.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalNonmonomial( Sl23 );  IsMinimalNonmonomial( S4 );</span>
true
false
</pre></div>

<p><a id="X7B416BBD80072079" name="X7B416BBD80072079"></a></p>

<h5>75.5-2 MinimalNonmonomialGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MinimalNonmonomialGroup</code>( <var class="Arg">p</var>, <var class="Arg">factsize</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a solvable minimal nonmonomial group described by the parameters <var class="Arg">factsize</var> and <var class="Arg">p</var> if such a group exists, and <code class="keyw">false</code> otherwise.</p>

<p>Suppose that the required group <span class="SimpleMath">K</span> exists. Then <var class="Arg">factsize</var> is the size of the Fitting factor <span class="SimpleMath">K / F(K)</span>, and this value is 4, 8, an odd prime, twice an odd prime, or four times an odd prime. In the case that <var class="Arg">factsize</var> is twice an odd prime, the centre <span class="SimpleMath">Z(K)</span> is cyclic of order <span class="SimpleMath">2^{<var class="Arg">p</var>+1}</span>. In all other cases <var class="Arg">p</var> is the (unique) prime that divides the order of <span class="SimpleMath">F(K)</span>.</p>

<p>The solvable minimal nonmonomial groups were classified by van der Waall, see <a href="chapBib.html#biBvdW76">[vdW76]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNonmonomialGroup(  2,  3 ); # the group SL(2,3)</span>
2^(1+2):3
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNonmonomialGroup(  3,  4 );</span>
3^(1+2):4
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNonmonomialGroup(  5,  8 );</span>
5^(1+2):Q8
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNonmonomialGroup( 13, 12 );</span>
13^(1+2):2.D6
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNonmonomialGroup(  1, 14 );</span>
2^(1+6):D14
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNonmonomialGroup(  2, 14 );</span>
(2^(1+6)Y4):D14
</pre></div>


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