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products/sources/formale Sprachen/GAP/pkg/recog/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 22.0.2025 mit Größe 63 kB

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<td class="tdleft">recognises large-base primitive permutation groups</td>
<td class="tdleft"><a href="chap6.html#X836212B6818B60FE">6.2-4</a></td>
</tr>
<tr>
<td class
< class" "StabilizerChainPermt
<td class="tdleft">
<td=""><a="chap6.#X7DC0B08B7C73A4F9"<="">20><a></td
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<td to input < class>/> is giantthat <pan=""></> orS_n> itsaction <span="">n<span). outputis structure class> a anSLP forarbitrary elementsof fromthe nice ;or< class=eyw>NeverApplicable rg/varis transitive code="keyw"></>in that wasthat =ArgG/> , was construction< class">wasunsuccessful Ifthe methodconstructs SimpleMathfor a permutation group using a stabilizer chain</td> <td class="tdleft"><a href="chap6.html#X834E9DFF7DC949DE">6.2-9</a></td> </tr /> </div> <a id="X80320AF97EF01284This methodis usedfor recognizing groups. <h5>6.2-1 This method is unused! This method <a id="X7F46D8727850585D" name="X7F46D8727850585D"></a> <h5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 This method is used for recognizing permutation groups. <>The triesto whetherthe <var""G/>is ( isspan=SimpleMath></spanor classSimpleMath>S_n innaturalonclass">n/span>points). output is either a data structure and aprocedure to write an SLPfor elementsof ">/>if ="><>istransitive <codeclass"fail/> thecasethatnoevidencewas < =Arg> or wasfound the span class"">D/span> unsuccessful.Ifthe constructs span ="SimpleMath/>thennode .<> <a id="X8757E1C37F03A49D" name="X8757E1C37F03A49D"></a> <h5>6.2-3 <code class="code">Imprimitive</code></h5> This method is used for recognizing permutation groups If the input group is not known to be transitive then this method returns <code class="keyw">NotEnoughInformation</code>. If the input group is known to be transitive and primitive then the method returns <code class="keyw">NeverApplicable</code>; otherwise, the If the method is successful then it also gives a hint for the children <pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <h5>6.2-4 <code class="code">LargeBasePrimitive</code></h5> This method is used for recognizing permutation groups. method todetermine the classGv> is fixed-point-free primitive that is nor alternating in natural action. This method is an implementation of <a href="chapBib.html#biBLNPS06">[LNPS06]</a>. <pAprimitive < classSimpleMathH/panon< =""Nspan is emlargeemif <span=SimpleMath/>,< class">r<spanwithspanclass=SimpleMathN={n k}r/span,and a isomorphism of spanclass""A_n ^ rS_n> and< classSimpleMath></spanact on <SimpleMathk/>- of<spanclassSimpleMath>1 ..,n ⋅ r ⋅ k^2 ≤ n</pan> Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 If <var class="Arg">G</var> is imprimitive then the output is <code class="keyw">NeverApplicable</code>. If <a id="X804A913E84B77F97" name>6.7< class class></code/pan <h5>6.2-5 <code class="code" This method is used for recognizing If If <a id="X83C651E58239EEAF" name="X83C651E58239EEAF"></a> <h5>6.2-6 <code class="code">NonTransitive</code><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 This method is used for <> a <var><var nontransitively method a to of classG/> on orbit largest. <var=""<var methodclass></code/ < <h5>6.2-7 <code class="code">Pcgs</code></h5> This method is used for recognizing permutation groups. This is the GAP library function to compute a stabiliser <a id="X82B9AF9F7A904FE5" name="X82B9AF9F7A904FE5"></apTODO <h5>6.2-8 <code class="code">PcgsForBlocks</code></h5 This method is unused! This method is called afterThis a homomorphism permutationvar"> to the action on moved points of>/> <var=Argvar any , is to or ratio pointsto moved acertain . If classG/var fixed butnot, then returnscode=""> that be again a later. In cases returns =""><code<pjava.lang.StringIndexOutOfBoundsException: Index 563 out of bounds for length 563 <a id="X834E9DFF7DC949DE" name="X834E9DFF7DC949DE"></a> <h5>6.2-9 <code class="code">StabChain</code></h5> <>his sused permutation/> This is the randomized aidjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62 <a id="X7DC0B08B7C73A4F9" name="X7DC0B08B7C73A4F9"></a> <h521 span=">codeclass"">StabilizerChainPerm/> This method is used for recognizing permutation groups. TODO <a id="X8236A4817FA037E2" name="X8236A4817FA037E2"></a> <h5>6.2-11 <code class="code">ThrowAwayFixedPointstd="> through generators and compare to the identity/java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73 This method is used for recognizing permutation groups> Thisclass="tdleft"><code="code">KnownStabilizerChain/code>/> <> current is if number class>/> of points at < class"13span of is at half of the number of points on which Arg><var < class"GAP/>. < fact methodcode=" fixed pointsbutdoes primitive,is efficienthandling .4./p> <a id="X829BA50B82FEC109" name=> << class< class</><td The following table gives an overview over the installed methods and their rank =""> =".#">span"63/ <div class="pcenter"><table class="GAPDocTable"> <caption class="GAPDocTable">Table: class="tdleft"><a href="chap6.html#X811FC944788D98396-<s>a>/> <tr> <td class="tdleft">3100</td> <td class class"90/> <td class="tdleft">go through <td class="tdleft"><code class="code"<code/> <td="tdleft"><a href".html#X8489BECB78664847"< classRefLink14/></>t> </tr> <tr> <td class="tdleft">1175</td> <td class="tdleft"><code class="code">KnownStabilizerChain</code></td> <td class="tdleft">use an already known stabilizer chain for this> <td class="tdleft"><a << ="X857BF1317CAAF260" ="">/>p </tr> <tr> <td class="tdleft">1100</td> <tdp>This method unused</> <td class="tdleft">check <td class=tdleftchap6#">< class="">6.3- </tr> <tr> <td class="tdleft">1000</td> <td class="tdleft"><code class="code">ReducibleIso</code></td> < class""> MeatAxe invariant/> <td class="tdleft"><a href="chap6.html#X811FC944788D9839">6.3-9<><a ="X81F74235823352B5"name"/<> <tr <tr> <td class="tdleft">900</td> java.lang.StringIndexOutOfBoundsException: Range [93, 94) out of bounds for length 62 < =" <td class="tdleft"><a href="chap6.html#X7A1D99F67E62B1D7">p< ="" ="">a </table> </div> <a id="X857BF1317CAAF260" name="X857BF1317CAAF260"></a> <h5>6.3-1 pThis used matrix/java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57 This method is unused! This method is only called when a hint was passedjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 The <5>6.-5 span="Heading"< classcode></code>/span<h5 <can used a method a projectivegroup <var=Arg<varIn case recognition has <code class"">.</code bound <codeclass></code and informationis down image kernel <a id="X81F74235823352B5" name="X81F74235823352B5"></a> <h5<a ="" ="B2C182B7898BC45"<aThis method is unused! <ppThisis recognizing matrix groups. < chain alreadyknownthen kernel given about stabilizer, the is use methods database . If chain node known for computation a chain node chain theway abovep <a id="X8795A42285A332BB" name="X8795A42285A332BB"></a> <h5>6.3-3 This method is unused! <a id="X7A9F8CA181F6412D" name="X7A9F8CA181F6412D"></a> <>6- <span=Heading class=">iagonalMatrices/code> This method is used for recognizing matrix groups This method is successful if and only if all generators of a matrix <a id="X7A1D99F67E62B1D7" name="X7A1D99F67E62B1D7"></a> <h5>6.3-5 <code class="code">GoProjective</code>6.39 <pan class"< class""ReducibleIso/>/5 This method is used for recognizing This method defines a homomorphism from a matrix group <var class="Arg">G</var> into the projective group <var class="Arg">G</var> modulo scalar matrices. In <a id="X7B2C182B7898BC45" name="X7B2C182B7898BC45"></a> <h5> This method is used for recognizing matrix groups. If a <a id="X84E09FCF8527D74A" name="X84E09FCF8527D74A"></a> <h5>6.3-7<h5>6.3-7 This method is unused!</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 This method is only called by a hint from <a href="chap6.html#X81F74235823352B5"><a id="X8007B3DC787D192C" name="X8007B3DC787D192C"></a> <h5>6.3-8 <code class="code">NaturalSL</code></h5> This method is unused! TODO <a id="X811FC944788D9839" name="X811FC944788D9839

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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href=< class>span="tocline>< =""> />.html">61 span class="">Methods generic groups/>

<a href="chap6_mj.html">[MathJax on]</a>
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<div class="ChapSects"><a href="chap6.html#X78C5295B86AD0C66">6 Methods for recognition</a>
<div class="ContSect"> <a href="chap6.html#X79A20FD17F834C70">6.1 <span

<div class="ContSSBlock">
 <a href="chap6.html#X78797CF584658CEC">6.1-1 <code class="code">FewGensAbelian</code></a>

 <a href="chap6.html#X7E5E93027A194885">6.1-2 <code class="code">KnownNilpotent</code></a>

 <a href="chap6.html#X878C1CED8191506C">6.1-3 <code class="code">SnAnUnknownDegree</code></a>

 >
/>
</div></div>
<div>
 /< classnocssnbsp;nbsps><a href=".html#">.-2< classHeading class">Giant/code><>
<div class="ContSSBlock">
 <a/>

 <a href="chap6.html#X7F46D8727850585D">6.2-2 

<<>
nbsp&;/spana "hap6.#X83C651E58239EEAF626<=Heading =">
 />
 />

 <a href="chap6.html#X84C3750C7A4EEC34">6.2-7 <code class="code">Pcgs</code></a>

 <a href="chap6.html#X82B9AF9F7A904FE5">6.2-8 <code class="code">PcgsForBlocks</code></a>
java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7
 <a href="chap6.html#X834E9DFF7DC949DE">/>
/>
 <a href="chap6.html#X7DC0B08B7C73A4F9">6.2-10 <code class="code">StabilizerChainPerm</code></a>

 <a href="chap6.html#X8236A4817FA037E2">6.2-11 <code class="code">ThrowAwayFixedPoints</code></a>

</div></div>
<div class="ContSect"> <a href="chap6.html#X829BA50B82FEC109">6.3 Methods for>

<div class="ContSSBlock">
 <a href="chap6.html#X857BF1317CAAF260">6.3-1 <code class="code">BlockDiagonal</code></a>

 <a href="chap6.html#X81F74235823352B5">6.3-2 <code class="code">BlockLowerTriangular</code></a>

 <a href="chap6.html#X8795A42285A332BB">6.3-3 <code class="code">BlockScalar</code></a>

 <a href="chap6.html#X7A9F8CA181F6412D">6.3-4 <code class="code">DiagonalMatrices/>

 <a href="chap6.html#X7A1D99F67E62B1D7">6.3-5 <code class="code">GoProjective>

 <a href="chap6.html#X7B2C182B7898BC45">6.3-6 <code class="code">KnownStabilizerChain</code></a>

 <a href="chap6.html#X84E09FCF8527D74A">6.3-7 <code class="code">LowerLeftPGroup</code></a>

 <a href="chap6.html#X8007B3DC787D192C">6.3-8 <code class="code">NaturalSL</code></a>

 <a href="chap6.html#X811FC944788D9839">6.3-9 <code class="code">ReducibleIso/span>

 <a href="chap6.html#X8635737086D48AE5">6.3-10 <code class="code">Scalar</code></a>

</div></div>
<div class="ContSect"> <a href="chap6.html#X867550F684F795AE">6.4 Methods for projective groups</a>

<div class="ContSSBlock">
 <a href="chap6.html#X87A53FB6813E2E3C">6.4-1 <code class="code">AltSymBBByDegree</code></a>

 <a href="chap6.html#X7F58277E780EEA2E">6.4-2 <code class="code">BiggerScalarsOnly</code></a>

 <a href="chap6.html#X863E0A1D7D7FEE04">6.4-3 <code class="/>

 <a href="chap6.html#X84FE699F85371643">6.4-4 <code class="code">Blocks</code></a>

 
 <a href="chap6.html#X8605DA5E85DFACDC">6.4-6 <code class="code">BlocksModScalars</code></a>

 <a href="chap6.html#X799F4A7A8239D70F">6.4-7 <code class="code">C3C5</code></a>

 <a href="chap6.html#X7E484B8F83849B49">6.4-8 <code class="code">C6</code></a>

 <a href="chap6.html#X7CE5D3D5876DE816">6.4-9 <code class="code">ClassicalNatural</code></a>

 <a href="chap6.html#X7EC8D0FD7DC6B213">6.4-10 <code class="code">ComputeSimpleSocle</code></a>

 <a href="chap6.html#X81E155F98671824A">6.4-11 
 <a href="chap6.html#X878ADA8A7CCE93E0">6.4-12 <code class="code">DoBaseChangeForBlocks</code></a>

 <a href="chap6.html#X7ED3241F84A98292">6.4-13 <code class="code">FindElmOfEvenNormal</code></a>

< < class&;&;<span=html>41 span=>code</><span/java.lang.StringIndexOutOfBoundsException: Range [187, 188) out of bounds for length 187

 <a href="chap6.html#X8634C79E7DB22934">6.4-15 <code class="code">KroneckerProduct</code></a>

 <a href="chap6.html#X865D3A1B7B6BB49B">6.4-16 <code class="code"code/>
/>


 <a href="chap6.html#X82B704EF80AE2FBB">6.4-18 <code class="code">NameSporadic/span

 <a href="chap6.html#X83E6AD6583DB7080">6.4-19 <code class="code">NotAbsolutelyIrred</code/>


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The following table gives an overview over the installed methods and their rank (higher rank means higher priority,

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<td class="tdleft">1250</td>
<td class="tdleft"><code class="code">FewGensAbelian</code></td>
<td="tdleft> few generators IsAbelianand yes,
<td class="tdleft"><a href="chap6.html#X78797CF584658CEC">6.1-1<d ="tdleft"< ".#X8236A4817FA037E2"< class>.<spana<td
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<td=""150/>
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<td class="tdlefttdclass=tdleft">97<td
<tr
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< class>0<td
<td class="tdleft"><code class="code">TrivialGroup</code></td>
<t>
<td class="tdleft"><a href="chap6.html#X8489BECB78664847">6.1-4</a></td>
</tr>
<tr>
<td class="tdleft">100</td>
<td ="tdleft"< classcode></></td
<td class="tdleft">try to find a huge amount of (possible internal) fixed points="">< =".htmlX804A913E84B77F97"< class.-<><a<>
<td class="tdleft>
</tr>
<tr>
<td class="tdleft">99</td>
<td class="tdleft"><code class="code">FewGensAbelian</code></td>
<td class="tdleft">if very few generators, check IsAbelian and if yes, do KnownNilpotent</td>
<td class="tdleft"><a href="chap6.
</tr>
<tr>
<<d class"dleft"80/td
<td class="td class="tdleftclass"/>td
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</tr>
<tr>
<td class="tdleft">95</td>
<td class="tdleft"><code class="code">MovesOnlySmallPoints</code></td>
<td class="tdleft">calculate a stabilizer>
<td class="tdleft"><a href="chap6.html#X804A913E84B77F97">6.2-5</a></td>
</tr>
<tr
<td class="tdleft">9td="tdleft"<a href"chap6html#X836212B6818B60FE"></a></djava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
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p this implemented suchthat canbe as forprojective < class><var that case node the class=code!projective<code component bound to <code ="keyw>rue/> this informationis passed down and kernel

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<h5> ="<=".#848D7F193ERefLink2>/>

This method is unused!

TODO

<a id="X867550F684F795AE" name="X867550F684F795AE"></a>

> span" projective

The =""> over with </>

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable">Table: Projective group find homomorphism methods</caption>
<tr>
class0>
<td class="tdleft"><code class=td="tdleft">=".html#">span=RefLinkspan<td
<td class="tdleft">go through generators and compare to the identity</td>
<td class="tdleft"><a hreftd="tdleft>00
<>
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<td class="tdleft"><code class="code">ProjDeterminant</tr
< =">00td>
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<tr
<tr>
<td class="tdleft">1200</td>
<td class="tdleft"><code class="code">ReducibleIso</code></td>
<td="tdleft">se MeatAxeto invariant</td
<dclass>a hrefhtml>spanRefLink.3<span<>
</tr>
<tr>
<td class="tdleft">1100</td
<td class="tdleft"><>
<td class="tdleft">write over a bigger field with smaller degree</td>
<td class="tdleft"><a href="chap6.html#X83E6AD6583DB7080">6.4-19</a></td>
</tr>
<tr>
<td class="tdleft">1050</td>
<td class="tdleft"><code class="code">ClassicalNatural</code></td>
< classtdleft whether isclassical in naturalrepresentationtd
<td class="tdleft"><a href="chap6.html#X7CE5D3D5876DE816">6.4-9</a></td>
</tr>
<tr>
<td class="tdleft">1000</td>
<td class="tdleft"><code class="code">Subfield</code></td>
<td class="tdleft">write over a smaller field with same degree</td>
<td class="tdleft"><a href="chap6.html#X7FE1FA217A08DCE5">6.4-24</a></td>
</tr>
<tr>
<td class="tdleft">900</td>
<td class="tdleft"><code class="code">C3C5tr
<td class="tdleft">compute a normal subgroup of derived and resolve C3 and C5</td>
<td class="tdleft"><a href="chap6.html#X799F4A7A8239D70F">6.4-7</a></td>
</tr>
<tr>
<td class="/
<td class="tdleft"><code class="code">C6</code></tdtd="tdleft"0java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
t=tdleft".X810B7F36818D626B>span ="RefLink41/>
<td class="tdleft"><a href="chap6.html#X7E484B8F83849B49">6.4-8</a></td>
</tr>
<tr>
<td class="tdleft">840</td ="tdleft"<class">ameSporadic/djava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62



<td class="tdleft"><code class="code">D247</code></td>

<td class

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>

<tr>

<td class="tdleft">810</td>

<td class="tdleft"><code class="code">AltSymBBByDegree</code></td>

<td class="tdleft">try BB recognition for dim+1 and/or dim+2 if sensible</td>

<td class="tdleft"><a href="chap6.html#X87A53FB6813E2E3C"><span class="

</tr>
<tr>
<td="">80/>
<td class="tdleft"><code class="code">TensorDecomposable</code></td>
<td class="tdleft">find a tensor decomposition</td>
<td class="tdleft"><a href="chap6.html#X7DE8D77D7D1E19E3">6.4-25<>
</tr>
<tr
<td class="tdleft">700</td>
<td class="tdleft"><code class="code">FindElmOfEvenNormal</code></td>
td="">find, or by of even subgroup>
<td class="tdleft"><a href="chap6.html#X7ED3241F84A98292">6.4-13</a></td>
</tr>
<tr>
<td classtdleft0<td
<td class="tdleft"><code class="code">LowIndex</code></td>
<tdtd=""> resort a stabilizerchain(rojectively<td
<td class="tdleft"><a href="chap6.html#X810B7F36818D626B"><spanclass"RefLink">6.4-<s></a>td
<//>br
<>
<td class="tdleft">580</td>
<td class="tdleft"><code class="code">NameSporadic</code></td>
<td class="tdleft">generate maximal orders</td>
<td class="tdleft"><a href="chap6.html#X82B704EF80AE2FBB">6.4-18</a></td>
</tr>
<tr>
<td class="tdleft">550</td>
<tdThismethod a black constructive (? recognition alternating symmetric.
<td class="tdleft">compute simple socle of almost simple group</td>
<td class="tdleft"><a href="chap6.html#X7EC8D0FD7DC6B213">6.4-10</a></td>
</tr>
<tr>
<td class="tdleft">500</td>
< classtdleft<code class="code">ThreeLargeElOrders</code></td>
<td class="tdleft">recognise Lie type groups and get its characteristic</td>
<td class="tdleft"><a href="chap6.html#X7A82C26E78FEF78F">6.4-26</a></td
</tr>
<tr>
<td
<td class="tdleft"><code class="code">LieTypeNonConstr</code></td>
=" recognition of groups/>
<td class="tdleft"><a href="chap6.html#X865D3A1B7B6BB49B">
<tr>
<td class="tdleft">100</td>
<td class="tdleft"><code class="code">StabilizerChainProj</code></td>
<td class="tdleft">last resort: compute a stabilizer chain (projectively)</td>
<td class="tdleft"><a href="chap6.html#X7A46A44F867CFB21">6.4-23<
</tr
</table> 
</div>

<a id="X87A53FB6813E2E3C" name="X87A53FB6813E2E3C"></a>

<h564-1 <span="Heading"><code class=code"AltSymBBByDegree

This method is used for recognizing projective groups.

This method is a black box constructive (?) recognition of alternating and symmetric groups.

This algorithm is probably based on the paper <a href="chapBib.html#biBBLGN+05">[BLN+05]</a>.

<a id="X7F58277E780EEA2E" name="X7F58277E780EEA2E"></a>

<h5>6.4-2 <code class="code">BiggerScalarsOnly</code></h5>

This method is unused!

TODO

<a id="X863E0A1D7D7FEE04" name="X863E0A1D7D7FEE04"></a></p< ="" =""<>

<h5>6.4-3 <code class="code">BlockScalarProj</code></h5

This method is unused!

This method is only called by a hintjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<a id="X84FE699F85371643" name="X84FE699F85371643"></a>

<h5>6.4-4 <code class="code">Blocks</code></h5>

This method is unused!

TODO

<a id="X84865B2E87FB00CE" name="X84865B2E87FB00CE"></a>

<h5>6.4-5 <code class="code">BlocksBackToMats</code></h5>

This method is unused!

TODO>

<a id="X8605DA5E85DFACDC" name="X8605DA5E85DFACDC"></a>

<h5>6.4-6 <code class="code">BlocksModScalars</code></h5>

This method is unused!

This method is only called when hinted from above. In this method it is understood that G should neither be recognised as a matrix group nor as a projective

This method does the balanced tree approach mapping to subsets of>6.49 < class=Heading"< class"">lassicalNatural/>

<a id="X799F4A7A8239D70F" name="X799F4A7A8239D70F"></a>

<h5>6.4-7 <code class="code">C3C5</code></h5>

This method is used<h5>64-0s class =></>>>

TODO</ethod used for projective.

<a id="X7E484B8F83849B49"<> methodrandomly the simple socle stores with information it forjava.lang.StringIndexOutOfBoundsException: Range [139, 133) out of bounds for length 328

<h5>6.4-8 <code class="code">C6</code></h5>

This method is used for recognizing projective groups.

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

<a id="X7CE5D3D5876DE816" name="X7CE5D3D5876DE816"></a>

<h5>6.4-9 <code class="code">ClassicalNatural</code>code<spanh5>

This method is used for recognizing projective groups.

TODO

<a id="X7EC8D0FD7DC6B213" name="X7EC8D0FD7DC6B213"></a>

<h5>6.4-10 <code class="code">FindElmOfEvenNormal/code></panh5>

 method usedfor projective.

This method randomly computes the

<a id="X81E155F98671824A" name="X81E155F98671824A"></a>

<h5>6.4-11 <code class="code">D247</code></h5>

This method is used for recognizing projective groups.

TODO

<a id="X878ADA8A7CCE93E0" name="X878ADA8A7CCE93E0"></a>

<5>.412 <spanclass="Heading"><codeclasscode>DoBaseChangeForBlocks/code></span/h5>

<pThisunused>

TODO

<a id="X7ED3241F84A98292" name="X7ED3241F84A98292"></a>

<h5>6.4-13 <code class="code">FindElmOfEvenNormal</code></h5>

This method is used for recognizing projective groups.

TODO

<a id="X7845D88F845EC143" name="X7845D88F845EC143"></a>

<h5>6.4-14 <code class="code">KroneckerKernel</code></h5>

This method is unused!

TODO

a id=" =X8634C79E7DB22934>6.4-15 <code class="code">KroneckerProduct</code></h5>

>!p

TODO

<a id="X865D3A1B7B6BB49B" name="X865D3A1B7B6BB49B"></a>

<h5>6.4-16 <code class="code">LieTypeNonConstr</code></h5>

This method is used for recognizing projective groups.

ml[<a>./p>

<a id="X810B7F36818D626B" name="X810B7F36818D626B"></a>

<h5>6.4-17 <code class="code">LowIndex</code></h5>

This method is used for recognizing projective groups.

This method is designed for the handling of the Aschbacher class C2 (stabiliser of a decomposition of the underlying vector space), but may succeed on other types<>The definesahomomorphism aprojective groupvarclass"G<var< =SimpleMath> PGL(d,)<>tothecyclicgroup span class==SimpleMath>()^/ ofspan="SimpleMath>d</>th in . The a group element <span=""> ∈ class=Arg>G/> determinant representativespanclass="<span,modulo<spanclassSimpleMath"></>

<a id="X82B704EF80AE2FBB" name="X82B704EF80AE2FBB"></a>

<h5>6.4-18 <code class="code">NameSporadic</code>

This method is used for recognizing projective groups.

This method returns a list of sporadic simple groups that the group underlying <var class="Arg">ri</var> could be. It does not recognise extensions of sporadic simple groups nor the Monster and the Baby Monster group. It is based on the Magma v2.24.10 function <code

<a id="X83E6AD6583DB7080" name="X83E6AD6583DB7080"></a>

<h5>6.4-19 <code class="code">NotAbsolutelyIrred</code></h5>

<pThis method used for If an irreducible projective group <var class="Arg">G</var> acts absolutely irreducibly then this method returns <code class="keyw">NeverApplicable</code>. If <var class="Arg">G</var> is not absolutely irreducible then a homomorphism into a smaller dimensional representation over an extension field is defined. A hint java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<a id="X850864887D7F193E" name="X850864887D7F193E"></a>

<h5>6.4-20 <code class="code">ProjDeterminant</code></h5>

This method is used for recognizing projective groups.

The method defines a homomorphism

<a id="X83288F2279C27632" name="X83288F2279C27632"></a>

<h5>6.4-21 <code class="code">PrototypeForC2C4</code></h5>

This method is unused!

TODO/FIXME: PrototypeForC2C4 is not used anywhere

<a id="X7B2111968598599F" name="X7B2111968598599F"></a>

<h5>6.4-22 <code class="code">SporadicsByOrders</code></h5>

This method is unused!

This method returns p>his method currently triesto find elementsto ofprimeorder Thisseems to quite well tensorfactors are not<></q>toomuchsuch enough ordersonbothtensor./>

<a

<h5>6.4-23 <code class="code">StabilizerChainProj</code></h5>

<> used forprojective./p>

This method computes a stabiliser chain and a base and strong generating set using projective actions. This is a last resort method since for bigger examples no short orbits can be found in the natural action. The h5642 spanclass""< =>/>

<>< id"X7FE1FA217A08DCE5=X7FE1FA217A08DCE5>/a>/p

<h5>6.4-24 <code class="code">Subfield</code></h5>

This method is used for recognizing projective groups.

TODO

<a id="X7DE8D77D7D1E19E3" name="X7DE8D77D7D1E19E3"></a>

<h5>6.4-25 <code class="code">TensorDecomposable</code></h5>

This method is used for recognizing projective groups.

TODO/FIXME: it is unclear if the following description actually belongs to this method, so be cautious!

This method currently tries to find one tensor factor by powering up commutators of random elements to elements of prime order. This seems to work quite well provided

This method and its description needs some improvement.

<a id="X7A82C26E78FEF78F" name="X7A82C26E78FEF78F"></a>

<h5>6.4-26 <code class="code">ThreeLargeElOrders</code></h5>

This method is used for recognizing projective groups.

In the case when the input group <var class="Arg">G</var>≤ PGL(d,p^e)<td=tdleft"<code class">nAnUnknownDegreecode<tdjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

This method assumes that its input is neither alternating nor sporadic and that <code class="func">ComputeSimpleSocle</code> (<a href="chap6.html#X7EC8D0FD7DC6B213">6.4-10</a>) has already been called.

This recognition method is based on the paper <a href="chapBib.html#biBKS09">[KS09]</a>.

<a id="X7D4A5A9A8266A292" name="X7D4A5A9A8266A292"></a>

<h4>6.5 Unused methods</h4>

The following table gives an overview overMethodsPerm</code>/td>

<div class="pcenter"><table class="GAPDocTable">
<<
<tr>
<td class="tdleft"><code class="code">KnownNilpotent</code></td>
<td class="tdleft"><code class="code">FindHomMethodsGeneric</code></td>
<td class="tdleft"><a href="chap6.html#X7E5E93027A194885">6.1-2</a></td>
</tr>
<tr>
<td class="tdleft"><code class=code"FindHomMethodsMatrix>/>
<td class="tdleft"><code class="code">FindHomMethodsGeneric<dclass=tdleft>ahref"chap6.#X8007B3DC787D192C">.38/><a>/>
<td class="tdleft"><a href="chap6.html#X878C1CED8191506C">6.1-3</a></td>
</tr>
<tr>
td="< ="ode></<td
<td class="tdleft"><code tdclass""< href=chap6html>
<td class="tdleft"><a href="chap6.html#X82B9AF9F7A904FE5">6.2-8</a></td>
</tr>
<tr>
<td =tdleft"<code ="ode></code<td>
<td class="tdleft"><code class="code">FindHomMethodsPerm</code></td>
<td class="tdleft"><a href="chap6.html#X80320AF97EF01284">6.2-1</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlockScalar</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="java.lang.StringIndexOutOfBoundsException: Range [70, 36) out of bounds for length 70
</tr>
<tr>
<td class="tdleft"><code class="code">NaturalSL</code></td>
<td class="tdleft"><td "tdleft>a>6-1/pan<a>td
<td class="tdleft"><a href="chap6.html#X8007B3DC787D192C">6.3-8</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">Scalar< ""<=#>< ="java.lang.StringIndexOutOfBoundsException: Range [81, 79) out of bounds for length 101
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.td class="tdleft"><code class="code"/<td
</tr>
<tr>
<td class="tdleft"><code class="code">BlockLowerTriangular</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.html#X81F74235823352B5">6.3-2</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlockDiagonal</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.html#X857BF1317CAAF260">6.3-1</a></td>
<tr>
tr
="< ="code>LowerLeftPGroup</code></d>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.html#X84E09FCF8527D74A">6.3-7</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">DoBaseChangeForBlocks</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X878ADA8A7CCE93E0">6.4-12</a></td>
</tr>
<tr>
<td class="< ""ahref=chtmlX83288F2279C27632>spanclass"">642<span<a>/>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 5
<td class="tdleft"><a href="chap6.html#X84FE699F85371643">6.4-4</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlocksModScalars</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X8605DA5E85DFACDC"><<td class=tdleft"<code class=code></>/>
</tr>
<tr>
<td class="tdleft"><code class="code">BlocksBackToMats</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X84865B2E87FB00CE">6.4-5</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">KroneckerProduct</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</tr>
<tr>
<td class="tdleft"><code class="code">KroneckerKernel</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X7845D88F845EC143">6.4-14</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BiggerScalarsOnly</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X7F58277E780EEA2E">6.4-2</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">PrototypeForC2C4</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X83288F2279C27632">6.4-21</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">SporadicsByOrders</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X7B2111968598599F">6.4-22</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlockScalarProj</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X863E0A1D7D7FEE04">6.4-3</a></td>
</tr>
</table> 
</div>

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quality100%

bsp;its description needs some improvement.

<a id="X7A82C26E78FEF78F" name="X7A82C26E78FEF78F"></a><pThismethodisused for recognizing groups</java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

<>.-6< =Heading">codeclass"code">ThreeLargeElOrders</code><h5

This method is used for recognizing projective groupsp><a id="" name""<a

In the case when the input group <var class="Arg">G</var>≤ PGL(d,p^e)</

This method assumes that its input is neither alternating nor sporadic and that <code class="func">ComputeSimpleSocle</code> (<a href="chap6.html#X7EC8D0FD7DC6B213">6.4-10</a>) has already been called.

This recognition method is based on the paper <a href="chapBib.html#biBKS09">[KS09]</a>.

<a id="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0

<h4>6.5 Unused methods</h4>

The following table gives an overview over the methods which are currently unused.

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<tr>
<td class="tdleft"><code class="code">KnownNilpotent</code></td>
<td class="tdleft"><code class="code">FindHomMethodsGeneric</code></td>
<td class="tdleft"><a href="chap6.html#X7E5E93027A194885">6.1-2</a></td>
</tr>
<tr>
<td class="> class=code"></code>/td>
<td class="tdleft"><code class="code">FindHomMethodsGeneric</code></td>
<td class="tdleft"><a href="chap6.html#X878C1CED8191506C">6.1-3</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">PcgsForBlocks</code></td>
<td class="tdleft"><code class="code">FindHomMethodsPerm</code></td>
<td class="tdleft"><a href="chap6.html#X82B9AF9F7A904FE5">6.2-8</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BalTreeForBlocks</code></td>
/<td>
<td class="tdleft"><a href="chap6.html#X80320AF97EF01284">6.2-1</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlockScalar</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.html#X8795A42285A332BB">6.3-3</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">NaturalSL</code></td>
tdleft=""></code>td>
< ""< =chap6htmlX8007B3DC787D192C/<tdjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
</tr>
<tr>
<td class="tdleft"><code class="code">Scalar</code></td>
<td class="tdleft"><code class=<td class="tdleft>code class="ode"PcgsForBlockscode>/>
< =tdleft>a ="chap6.#X8635737086D48AE5">6.3-10</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlockLowerTriangular</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6class">< class"ode"BalTreeForBlocks>/td
</tr>
<tr>
<td class="tdleft"><code class="code">BlockDiagonal</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.html#X857BF1317CAAF260">6.3-1</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">LowerLeftPGroup</code></td>
<td class="tdleft"><code class="code">FindHomMethodsMatrix</code></td>
<td class="tdleft"><a href="chap6.html#X84E09FCF8527D74A">6.3-7</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">DoBaseChangeForBlocks</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"< href="chap6.html#X878ADA8A7CCE93E0">.41</pan></a>/td>
</tr>
<tr>
<td class="tdleft"><code class="code">Blocks</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<d class=tdleft>a href"chap6.htmlX84FE699F85371643"<panclass"efLink">6.4-4</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlocksModScalars</code></td>
<>FindHomMethodsProjective<code>/>
<td class="tdleft"><a href="chap6.html#X8605DA5E85DFACDC">6.4-6</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">BlocksBackToMats</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X84865B2E87FB00CE">6.4-5</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">KroneckerProduct</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X8634C79E7DB22934">6.4-15</a></td>
</tr>
<tr>
<tr
<td class="tdleft"><code<>
<td class="tdleft"><a href="chap6.html#X7845D88F845EC143">6.4-<tdclass=tdleft>code class"LowerLeftPGroupcode</d
</tr>
<tr>
<td class="tdleft"><code class="code">BiggerScalarsOnly</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X7F58277E780EEA2E">6.4-2</a></td>
</tr>
<tr>
<td class="tdleft"><code class="code">PrototypeForC2C4</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<d class=tdleft"< href="hap6.#X83288F2279C27632"< =RefLink>.-1/span>/a>td>
</tr>
<tr>
<td class="tdleft"><code class="code">SporadicsByOrders</code></td>
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X7B2111968598599F">6.4-22</a></td>
</tr>
<tr>
td=>code="">BlockScalarProjcode<td
<td class="tdleft"><code class="code">FindHomMethodsProjective</code></td>
<td class="tdleft"><a href="chap6.html#X863E0A1D7D7FEE04">6.4-3</a></td>
</tr>
</table> 
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