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java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 50
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< =c">span class=pan>

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<span class"ontSS"><br /><span class="nocss">  </span><a href="chap69_mj<spanclass"ContSS"<br>span class="nocss">  </span><a href="chap69_mj.html#X781772FD865B9F9C">69.6-2 MTX.span class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.html#X7F00E49484FBA7B8">69.6-3 MTX.HomogeneousComponents
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<span class="ContSS"> class="ContSS"><br >span class=nocss>&<span class="ContSS"><br /><span class="nocss"> a"html#"69.2 MTXIndecomposition
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<span</iv<div
<span class="ontSS><br /><span class="nocss" nbsp;/span<a href="chap69_mj.html#X7E84E1927EBFD483">69.5-3 MTX.DegreeSplittingField<pan class"ContSS"<br /><span class="nocss">  </span><a href="chap69_mj.html#java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 7
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</>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.html#X81A6ECB078D4441C">69.10-5 MTX.Distinguish</a></span>
<span class="ContSS"><brspan class="ContSS"><br /<span classbr>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 157
<class"ContSS">  /<span=">nbsp;n;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 146
<="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.html#X7F00E49484FBA7B8">69.6-3 MTX.span class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.html#X8168EB348474046B">69.11-5 MTX.OrthogonalSign</a><>
</div></div>
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 176
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
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<<panclass= class="ContSS"><br /><span class="nocss">  </span><a<span class="ContSS"><br span class="ContSS class=CntSS"br />pan =ocss"nbsp; </span><a href="chap69_mj.html#X8753A03A7C7CBFF1">69.8-4 MTX.InducedActionSubMatrix<<pan class="ContSS"<br/span ="ocss">  </span><a href="chap69_mj.html#X7E692DC97AFB661E">69java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 12
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.span class="ContSS"><br /><span class="nocss"> span ="ContSS">r  /span class="nocss">  </span><a java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 7
<class="ontSS" <java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7
<;</span><ahref"chap69_mj.#864527B77A359195">697-  /<span=>nbsp&;/pan< href=chap69_mj.html#X7D6C947A7C8C14B2"6913-3 SMTX..lgElMat
< class"nocss&;&bsp;span<href=hap69_mjhX830500CE7ABF6039">69.7-6 MTX.BasisRadical</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 141
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<span class="ContSS"><br /><span class="nocss">  </span><a ref="chap69_mj.html#X8367B4A17EC39ABD">69.13-6 SMTX.AlgElNullspaceVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.html#X7E5038F384DBCAEC">69.7-11 MTX. class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj.html#X81A6ECB078D4441C">69.10-5 MTX.Distinguish<div
</div></div>
div class="ContSect">span class="tocline"><span class="nocss"> </span><a href="chap69_mj.html#X7AE730FB81ED86FE">69.8 <span class=</span
<span
<div class="ContSSBlock">
<span classpanclass""br >pan"">;bsp<span< href=chap69_mj.html#X78E60EFE802AEBC1">69.11-4 MTX.BasisInOrbit</a></span>
<span class="ContSS">br/<< ="ContSS<br /<span class=nocss>&;bsp;<java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 146
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</div></div>
<div class="ContSect"><span class="toclinejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<div class="ContSSBlock"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<span">69.9-2 MTX.BasisModuleEndomorphisms<a</pan>
<pan ="">r/< class="nocss"> nbsp;</span><a href="chap69_mj.html#X8519B3C486AC8C7E">69.9-3 <p>creates a MeatAxe module over <var class="Arg">field</var> from a list of invertible matrices <var class="Arg">gens<a java.lang.StringIndexOutOfBoundsException: Range [147, 146) out of bounds for length 244
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 150
</div></div>
<div class
</span>
<div <java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 147
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<<class=""< ""width"00">tr> class="tdleft"<code class="">&#8227; GModuleByMatscode> <var class=Arg"mptygens</var> <var class="Arg">dim</<>creates a MeatAxe  varclass=<var>from   ofinvertible <p>Called with a permutation group <var class="Arg">G</var> and a field <var class="Arg">F</var> infinitefunc"PermutationGModulecode thenatural permutationmodule spanclass=impleMath"(M\span over var class=Arg>F<var for the group of permutation matrices that acts on the canonical basis of <span class="SimpleMath">\(M\)</span> in the same way as <var class="Arg">G</var> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ="> =nocss>nbspn;/
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<<.- <
</
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<class"< /<span n">java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 150
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap69_mj
<panclassbr/spanclass="nocss">  </span><a href="chap69_mj.html#X78A6B95686671067">69.13-8 SMTX.CentMat</a></span>
<span =ContSS"<br/>< class="nocss">  </span><a href="chap69_mj.html#X7D199DB6804F5D8F">69.13-9 SMTX.java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 0
</div></div<5>4-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</div>

<h3p> withagroup <p>returns the in which the matrices act.</p>

<

<><trong class=pkg"GAP/strong> uses the improved java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<divclass="func"><table class="func" width="100%"><tr><td class="tdleft">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X85B05BBA78ED7BE2

<h4>69.1 <span class="Heading">MeatAxe Modulesp>codeclass=fjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0

<>< id=X801022027B066497 pjava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 114

<h5>69.1-1 <span class="Heading">GModuleByMatsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<
<p><a=XC"7C77D22782C98D4E></a</p>
eldvara  of java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 0

<p>MeatAxe routines arethedegreeof splittingfield   theprimeMeatAxeroutinesaccessed theglobal <ode ""codew  a java.lang.StringIndexOutOfBoundsException: Range [110, 109) out of bounds for length 388


<dl>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>


<dd
<dt><strongjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<dd>pa=X78E61F7287BF1D0Cjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0

<dd>
<dt><strong div class="func"=">java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
dd>p> which  space is .<p

</dd>
<>whether<class"Arg"module</ar>  indecomposable</>
<p>h569.-

<p><a id="X87B82250801A1BD0" name="p>returns a decomposition of <var class="Arg/java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h4>69.2 <span p>returns the field which <varclass"rg>java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 289

<p><a id="X860E128B7D388FBE" name="X860E128B7D388FBE"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>69.2-1 <span class="Heading">NaturalGModule</java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 0
<p>creates a MeatAxe module over <<dd><p>one of the indecomposable components,</p

<"X8233134A81D58DA3"name=X8233134A81D58DA3>

<h5div class="">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func
<permutationclass

<p><a id="X809B2F4585E7E7A5" name="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>69.2-d =func"<table class="func" width="100%"><tr><td class=">A module is absolutely irreducible if it remains irreducible over the:  thejava.lang.StringIndexOutOfBoundsException: Range [125, 123) out of bounds for length 259

<"java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<field varclassArg"F</var

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<h5>69.5>69.-2MTXProperSubmoduleBasish5

=""id=D9B5B4E7F5A5FBD" java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 62
<6961 java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 36

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<p69.2 MTXjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 61

<

<p><a decomposition of<r class=module<var as adirectsumof indecomposablemodules. It returns a list, each entry is a list of form [<var java.lang.StringIndexOutOfBoundsException: Index 161 out of bounds for length 0

<p><a id="X7C2352A17B505AF6" name="X7C2352A17B505AF6"></a></<pp>returnsa list ofbases  <

<h5>69.3-1 MTX</h5>p><aid"BA359195" =XB77A359195<>p>

<div class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< hold the

<p>a ="84AB808B7C543377name="84AB808B7C543377"></a></p>

<>69. <span class="Heading">Accessing a Module</span></h4>

<p

<p><a id="X78E61F7287BF1D0C" name="<><strong class"Mark"><codejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>69.4-1 MTX.Generators</h5>

<div class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<><a id="X7DF2D6C07D7B09CD"name="DF2D6C07D7B09CD</</java.lang.StringIndexOutOfBoundsException: Range [1, 61) out of bounds for length 0

<h5>69.4

<div classjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X830C00887CE9323C" name="X830C00887CE9323C">

<h5>69.4-3MTX.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><preturnsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
pjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0

< id"X84D04C7E8423EB5D

<h469.. <span> class==Arg><var should be  subspace of or avector in) the underlying vector 

<" java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 62

java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 33

<div class="func"><table class=<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ptestsX84604D867983DD41=java.lang.StringIndexOutOfBoundsException: Range [56, 52) out of bounds for length 62

<<id="876810java.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for length 39

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class<d <=""%<java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 238
<p>A module is absolutely irreducible if it remains irreducible over the java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 0

<<

<h5>69.java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0

<div class="func"><table
<java.lang.StringIndexOutOfBoundsException: Range [14, 2) out of bounds for length 81

<p><a id="X791BA495829669C4" name="X791BA495829669C4"<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<696  <<

<p>A module

<p><p>< java.lang.StringIndexOutOfBoundsException: Range [10, 2) out of bounds for length 71

<696-1MTXh

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<h5>69.6-2 MTX.Indecomposition<

<div class="func"><table class="
<returns a decomposition of <var class="Arg">module</var> as a direct sum of indecomposable modules. It returns a list, each entry is 

<p><a id

<h5>69.6-3 MTX.HomogeneousComponents

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p> the   <var java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 289


<dl>
<dt><strong class="Mark"><code class="code">indices<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<dd><p>the index numbers  to the action  class=Ajava.lang.StringIndexOutOfBoundsException: Range [80, 78) out of bounds for length 449

</dd>
<dt> class=Mark><code  class=code<code></></>
<dd><p

</dd>
<dt><strong class="
<dd 

</
</dl>
<p><a id="X85A258567D96B9BE" name="X85A258567D96B9BE"></a></p>

<h4>69.7 <span class="Heading">Finding Submodulescreates  java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 0

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

 "<java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 237

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.SubmoduleGModulepjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 460
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code<div class="unc"><table =func"width""><><td class"tdleft>code class=func>#8227 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><var class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<aid" =X81326java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<>69.2 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 236
 submodule<java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 145

java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 298

<h5>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div h5>69.9-2thesamejava.lang.StringIndexOutOfBoundsException: Range [24, 20) out of bounds for length 100
<  class="func"><table idth"100>>class""<code class=">#8227;./ode( class"Arg">var<td>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X871D9AF87FABFB00" name="<p><a id="X8519B3C486AC8C7E" name="X8519B3C486AC8C7Eclass=t">< =func>& .ducedAction<code>< class">module/>, var classA"sub<var[ < =Arg"type/]<<p>Computes induced actions on submodules or factor modules and also returns the corresponding bases. The action taken is binary encoded in <var class="Arg">type</var>: <code class="code">1</code> stands for subspace acticode/code>for  the module    basisjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5><h5>699- BasisModuleHomomorphisms

<class"unc"<classfunc><ableclass=""idth"%">trclass"tdleft">codeclass=func>&#8227;MTX.</code(< ="Arg"module1<>/</tr>/>
<p a list ofof allminimal submodules.<p

<p><" name"X864527B77A359195">/a</p>

<h5

<divclass"unc><ble class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MTX.BasesMaximalSubmodules</code>( <var class="Arg">module</var> )</td><
<p>returns a list of bases of all maximal submodules.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X830500CE7ABF6039" java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0

<</h5>

< =functable "><td class=tdleft"<codeclass=unc>#8227.Equivalent/ode( <=">module1</>,var class="rg"></ar)/td><td class="tdright">( function )</td></tr></table></div>
<p id"

<p><a id="

<h5>69.7-7 MTX.BasisSocle

<div class="func"><table class="func" width
ocleof<>module<var./>

<p><a id="X7F7FB6687ADE3FD8<h469.10<span="java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0

<java.lang.StringIndexOutOfBoundsException: Range [33, 3) out of bounds for length 44

<<ivclass="
<p><var="<>java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 504

<p><a id="X79B704998400B9FC" name="java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0

<h5>69.7-9 MTX.BasesCompositionSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7E77F9A97EA855E2

<>697-0 returns a basisof the space of homomorphisms from the irreducible module <var class="Arg">module1</var> to <var class="Arg">module2</var>.</p>

< class="java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0

<p><a id="><var class="Argjava.lang.StringIndexOutOfBoundsException: Range [52, 36) out of bounds for length 504

<h5

< func>class"funcwidth"
<p>java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 101

<p><a id>a id="=java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<h4>9. <span class=H

<=java.lang.StringIndexOutOfBoundsException: Range [43, 36) out of bounds for length 237

<h5>69.8-1 MTX.NormedBasisAndBaseChange</h5>

<div class="func"><table class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0

>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<=
<div h5>69.11-1 MTX.java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>creates a new class"func"= 100"><tr<td classjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 176

<

<>.3MTX<h5>

<div class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
creates anewmodulecorrespondingtoaction java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 344

<p>The basis <var class="Arg">sub</var> has to be given in normed form. (That is it java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 0

<ph5>69java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>69.8-4 MTX.InducedActionSubMatrix</h5>

<div class="func"><table class="func"
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 265
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 298
>

<p>a="X7B137BE5877A7FA1java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 62

<h5>6985MTX class=GAPprompt;<span span=GAPinput> InvariantQuadraticForm ));/>

<div class=1 anclass""><span<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 94
< the corresponding.Theaction 

<p>All these routines return <code class="keyw">failspan class=>java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 83

<p><a id="X8040270F791514C8" name="X8040270F791514C8"></a></ ""gap&<G"gapt/java.lang.StringIndexOutOfBoundsException: Range [40, 37) out of bounds for length 94

<h4>69.9 <span class="fail

<p><a id="X8292535D8533671C=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 94

<h5>69.9-1 MTX<span class"pa78java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<div class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<returns ofthe

<

<h5>69.9-2 MTX.BasisModuleEndomorphisms</h5>

<div class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns a java.lang.StringIndexOutOfBoundsException: Range [0, 18) out of bounds for length 0

<8519 namejava.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<>69.3< ""table"width"%"< java.lang.StringIndexOutOfBoundsException: Range [79, 72) out of bounds for length 230

<div class="func"><table <var an irreducible spanclass=SimpleMath"\G)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="<p>The <codeclass="java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 108

<h5>69.9-4 MTX.ModuleAutomorphisms</h5>

<div div class="xample><pre>
<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 166

<p><a id="X850324FF7912A541" name="span class="GAPprompt">gap></span> <span">OrthogonalSignjava.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 120

<span class"APprompt"gap&;/pan><="&;/<class"java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 120

< lower-java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 2

<p><a idspan class=GAPprompt"gapgt;/spanclass="APprompt&;<pan < =GAPinput".rthogonalSign GModuleByMats    )/>

<h5>69.10-1 MTX.IsEquivalent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.IsEquivalent</code>( <var class=span class"GAPprompt"gapgt;/> < =">.OrthogonalSign(GModuleByMats(mats, GF(2) ) );</span>
<

<p><a id="h4>69.12 <span class="Heading

<>

<div java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<p>returns an isomorphism from <var class="Arg">module1</var> to <var java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 0

<3"name the module action on a random irreducible

<java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
sArg"mat<var> shouldvar class"Arg><ar  class=SimpleMath>(times\<span <varclass="<var> defining homomorphism fromclass=Arg"module1/>to <varclass"Arg

2860"</a<>java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 126

<</>

<div class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns a basis of the space of all homomorphisms

<p><aclass=<tablejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 293

<h5>69.10-5 MTX.Distinguish

<div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>div="func"<java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 298

<="X7B426E4679C1AF25" name="7B426E4679C1AF25B426E4679C1AF25"/></p

<h4>69.11 <span class="java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0

<p>The functions in this bGModulesh5

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

<h5>69.11-1 MTX.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><B5"name=></a>></p>
p

<p><a id

6911-./h5

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0

<p><a id="<div class="<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 223

<h5>69.11-3 MTX.InvariantQuadraticForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdlefteataxe.()code></>
p eitherthe of      the absolutelyirreduciblemodule<class""module<

<p>If the characteristic of <var class="Arg">module</var> is odd then <code class="keyw

 java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 92


<div class="
>gtjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 87
< <span
<span<h5>6912- java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
 . . . .
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 8
 . . 1 .
    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<span class="GAPprompt""79F16677C2EE095">/<>
<span java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 50
p>c java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 50
<span class="GAPprompt">gap<5>9.12-1SMTX.ompleteBasis/java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
pan=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 94
<span class="GAPprompt">gap></span> <span class="GAPinput">q:= MTX.p>extends the partial basis <var class="Arg">pbasis</var> to a basis of the full spaceArg/java.lang.StringIndexOutOfBoundsException: Range [0, 129) out of bounds for length 0
<class="GAPprompt">gap></span> <span class="GAPinput">q = - TransposedMat( q );  # antisymmetric inv. bilinear form</span>
true
</pre></div>

<p><a id=<>  getter  accessinternal  Foreachroutine, the  setters java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 155

java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0

<
<p>returns ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X8168EB348474046B" name

<h5>69.11-5 MTX.OrthogonalSign

<div class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
impleMath()/span- If<var ="module</var> does not fix a nondegenerate quadratic java.lang.StringIndexOutOfBoundsException: Index 187 out of bounds for length 102

<p>An error is signalled if <var class="Arg">module</var> is p>list <code class="code">[newgens,coefflist]</code> giving an used forchopping.</p>

<p>The code class="<>idX7D6C947A7C8C14B2"ame"D6C947A7C8C14B2">/a<p


<div class="example"><pre>
<span class="GAPprompt">gapajava.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
< class="APprompt>gap></span> <span class="GAPinput">MTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );</span>
1
<span class="GAPpromptpmatrixof <code class="unc>SMTX.AlgEl</code> ( href="chap69_mj.X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</p>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
-< id="417F86A7A20F128" "X8417F86A7A20F128"<a<p
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
0
<class">&t</>span class="GAPinput">mats:= f="chap69_mj.html#X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</p>
<span class="GAPprompt">gap></span> <span p<ajava.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0
fail
</>

<><a  id="B0E3237BA056FC name="X87B0E3237BA056FC"></a></p>

<h4>69.12 <span class="Heading">The Smash MeatAxe</span></h4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The standard MeatAxe provided in the <strong

<p>><a id"java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0

<h5>69.12-1 SMTX.RandomIrreducibleSubGModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<

pa =java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<.22 >  thenullspace/java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 34

<class"func<table class="func" width="100%"><tr><td class="tdleft"><code
<p>finds an java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0

<p><a id="X811339547D341BBE" namejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>matrix centralising all java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 ="java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 293
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 78

<p><a id="div java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 265

<h5>69.12-4 SMTX.MinimalSubGModules</h5>

<div class="func"><table class="func" width="100%< ="=chlink1"GotoChapter: </span><a ".>Top/a  <a href="ducibilityTestcode> (<href=chap69_mjhX7E692DC97AFB661E"><span class="RefLink">69.12-8</span></a>).</p>
Arg>ax/var<00%"<tr><td class="tdleft">code class"func">#8227; MTX.CentMatMinPoly</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>

<p><a id="X87E49FCD867983B5" name="X87E49FCD867983B5"></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>69.12-5 SMTX.Setter</h5>

<div class="func"><table class="func" width="100%"><tr><div class="chlinkbot"<class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns a setter function for the component <code class="code">smashMeataxe.(string)</code>.</p>

<p><a id="X7E60EBC57FFDF7BD" name="X7E60EBC57FFDF7BD"></a></p>

<h5>69.12-6 SMTX.Getter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.Getter</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a getter function for the component <code class="code">smashMeataxe.(string)</code>.</p>

<p><a id="X808345D784E0AC85" name="X808345D784E0AC85"></a></p>

<h5>69.12-7 SMTX.IrreducibilityTest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.IrreducibilityTest</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests for irreducibility and sets a subbasis if reducible. It neither sets an irreducibility flag, nor tests it. Thus the routine also can simply be called to obtain a random submodule.</p>

<p><a id="X7E692DC97AFB661E" name="X7E692DC97AFB661E"></a></p>

<h5>69.12-8 SMTX.AbsoluteIrreducibilityTest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AbsoluteIrreducibilityTest</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests for absolute irreducibility and sets splitting field degree. It neither sets an absolute irreducibility flag, nor tests it.</p>

<p><a id="X80BC392285994DA8" name="X80BC392285994DA8"></a></p>

<h5>69.12-9 SMTX.MinimalSubGModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.MinimalSubGModule</code>( <var class="Arg">module</var>, <var class="Arg">cf</var>, <var class="Arg">nr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the basis of a minimal submodule of <var class="Arg">module</var> containing the indicated composition factor. It assumes <code class="code">Distinguish</code> has been called already.</p>

<p><a id="X79EF16677C2EE095" name="X79EF16677C2EE095"></a></p>

<h5>69.12-10 SMTX.MatrixSum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.MatrixSum</code>( <var class="Arg">matrices1</var>, <var class="Arg">matrices2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates the direct sum of two matrix lists.</p>

<p><a id="X7D1471077A774C81" name="X7D1471077A774C81"></a></p>

<h5>69.12-11 SMTX.CompleteBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.CompleteBasis</code>( <var class="Arg">module</var>, <var class="Arg">pbasis</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>extends the partial basis <var class="Arg">pbasis</var> to a basis of the full space by action of <var class="Arg">module</var>. It returns whether it succeeded.</p>

<p><a id="X7FDF8F3F83B83336" name="X7FDF8F3F83B83336"></a></p>

<h4>69.13 <span class="Heading">Smash MeatAxe Flags</span></h4>

<p>The following getter routines access internal flags. For each routine, the appropriate setter's name is prefixed with <code class="code">Set</code>.</p>

<p><a id="X84A93AC482A1946D" name="X84A93AC482A1946D"></a></p>

<h5>69.13-1 SMTX.Subbasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.Subbasis</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Basis of a submodule.</p>

<p><a id="X7ABCD69880772B2D" name="X7ABCD69880772B2D"></a></p>

<h5>69.13-2 SMTX.AlgEl</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgEl</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>list <code class="code">[newgens,coefflist]</code> giving an algebra element used for chopping.</p>

<p><a id="X7D6C947A7C8C14B2" name="X7D6C947A7C8C14B2"></a></p>

<h5>69.13-3 SMTX.AlgElMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElMat</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>matrix of <code class="func">SMTX.AlgEl</code> (<a href="chap69_mj.html#X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</p>

<p><a id="X8417F86A7A20F128" name="X8417F86A7A20F128"></a></p>

<h5>69.13-4 SMTX.AlgElCharPol</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElCharPol</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>minimal polynomial of <code class="func">SMTX.AlgEl</code> (<a href="chap69_mj.html#X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</p>

<p><a id="X79A82FED785BFB6D" name="X79A82FED785BFB6D"></a></p>

<h5>69.13-5 SMTX.AlgElCharPolFac</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElCharPolFac</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>uses factor of <code class="func">SMTX.AlgEl</code> (<a href="chap69_mj.html#X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</p>

<p><a id="X8367B4A17EC39ABD" name="X8367B4A17EC39ABD"></a></p>

<h5>69.13-6 SMTX.AlgElNullspaceVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElNullspaceVec</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>nullspace of the matrix evaluated under this factor.</p>

<p><a id="X877F1AB77DC1E12C" name="X877F1AB77DC1E12C"></a></p>

<h5>69.13-7 SMTX.AlgElNullspaceDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElNullspaceDimension</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>dimension of the nullspace.</p>

<p><a id="X78A6B95686671067" name="X78A6B95686671067"></a></p>

<h5>69.13-8 SMTX.CentMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.CentMat</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>matrix centralising all generators which is computed as a byproduct of <code class="func">SMTX.AbsoluteIrreducibilityTest</code> (<a href="chap69_mj.html#X7E692DC97AFB661E"><span class="RefLink">69.12-8</span></a>).</p>

<p><a id="X7D199DB6804F5D8F" name="X7D199DB6804F5D8F"></a></p>

<h5>69.13-9 SMTX.CentMatMinPoly</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.CentMatMinPoly</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>minimal polynomial of <code class="func">SMTX.CentMat</code> (<a href="chap69_mj.html#X78A6B95686671067"><span class="RefLink">69.13-8</span></a>).</p>


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