<p>The java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 7
<><idX78797CF584658CECname="89CF584658CEC"<a<p>
<h5>6.1-1 <span class="Heading"><code class="code">java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<p>This method is forrecognizing permutation groups.</p>
<p>IfpThe methods be equally applied permutation,matrix projectivegroups do referto asblack-boxgroups here they allowed containcode onlyworks inputs thelisted typesp>
<h4>6.2 <span class="Heading">Methods for permutation groups</span></h4>
<p>The following table gives an overview over the installed methods and></span></h5
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Permutation group find homomorphism methods</caption>
<tr>
<tdclass"tdleft">1250/td
<td class="tdleft"><code class="code">FewGensAbelian</code></td>
<td=tdleftifvery generators, check andif, do</td
<td class="tdleft"><a href="chap6.html#X78797CF584658CEC"><span class="RefLink">6.1-1</span></a>
</tr>
<tr
< class150/td
<td classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 td="ifvery ,checkIsAbelian andif ,doKnownNilpotent/d>
<td>This methodissuccessfulif andonly if all generators of a group <varclass="Arg">G</var> areequalto identity. Otherwise itreturns <<codeclass="keyw">NeverApplicable</code indicating thatitwill never succeed This method onlyinstalled handle the casesuch that do have to this case account theother.</pjava.lang.StringIndexOutOfBoundsException: Index 363 out of bounds for length 363
<tr
<tr>
<td class="tdleft">300</td>
<td class="tdleft"><code class="code">TrivialGroup</code></td>
<td class="tdleft">go through generators
<tdclass"dleft"><a hrefchap6#X8489BECB78664847">span ="RefLink">6.1-
</tr
<tr>
<td class="tdleft">100</td>
<td class="tdleft"><code class="code">ThrowAwayFixedPoints</code></td>
<td class="tdleft">try to find a huge amount of (possible class"ifvery ,checkIsAbelian if ,doKnownNilpotent
<class">span="RefLink"6.2-11/></a>/>
</tr>
<tr>
<td class="tdleft">99</td>
<td class="tdleft"><code class="code">FewGensAbelian</code></td>
<td class"dleft">050td
< classtdleft>a ="chap6.#X78797CF584658CEC"< classRefLink.1-</span<a><td
</tr>
<tr
< =tdleft7</>
<td class="tdleft"><code class="code">Pcgs</code></>
<td class="tdleft"td="tdleft"30<>
<td class="tdleft"><a href="chap6.html#X84C3750C7A4EEC34"><span class="RefLink">6.2-7</span></a></td>
</tr>
<rjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<td class="tdleft">95</td>
<td class="tdleft"><code class="code">MovesOnlySmallPoints</code></td>
<td class="tdleft">calculate a stabilizer chain if only smalltdclass>code="">hrowAwayFixedPointscode/>
<td classtdleftahrefchap6#>span="RefLink">62-5/span<a><td
</tr>
<tr
<td class="tdleft">90</td>
<td class="tdleft"><code class="code">NonTransitive</code></td>
<td class="tdleft">try to find non-transitivity and restrict to orbit</td>
<td class="tdleft"><a href="chap6.html#X83C651E58239EEAF"><span class="RefLink">6.2-6</span></a></td>
/
<tr>
<=">
td">code"Giant/>
<td class="tdleft">tries<td class="tdleft"useaPcgs calculate a stabilizer chain</td>
<td class="tdleft"><a href="chap6.html#X7F46D8727850585D"><span class="RefLink">6.2-2</span></a></td>
</tr
<tr>
<td class="tdleft">70</td>
<td class="tdleft"><code class="code">Imprimitive</code></td>
<td class="tdleft">for a imprimitive permutation group, restricts to block system</td>
<td class="tdleft"><a href="chap6.html#X8757E1C37F03A49D"><span class="RefLink">6.2-3</span></a></td>
</tr>
<tr
<td class="tdleft">60</td>
<td class="tdleft"><code class="code">LargeBasePrimitive</code></td>
<td classtr>
< class> =.X836212B6818B60FE ="">.-<span/>
</tr>
<tr>
<tdclass5</td
<td class="tdlefttdlefttoand orbit "f a permutation group usingsinga stabilizer via the spanclass""> package><td
<td<>
</tr tr
< class"5/>
<td class="tdleft"><code class="code">StabChain</code></tdactions>
<td class="tdleft">for a permutation group using a stabilizer chain</td>
<td class="tdleft"><a href="chap6.
<tr
</table><br />>/>
</div>
<p>This method creates a balanced composition tree for the kernel of an imprimitive group. This is guaranteed as the method is just called from <code class="code">FindHomMethodsPerm.</code><a href="chap6.html#X8757E1C37F03A49D"><span class="RefLink"><span class="Heading"><code class="code">Imprimitive</code></span></java.lang.StringIndexOutOfBoundsException: Range [0, 322) out of bounds for length 74
<p><a idtd="tdleft>><code class classtdleft hrefhtml>span classRefLink>.-1</span/>
<p>This method is used forjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 4
<p>The method tries determine whethertheinput groupvar="Arg"></aris a giant ( is, span classSimpleMathA_nspanor <span class="SimpleMath"></span> in natural action on<span classSimpleMathn/> points. The eitheradata <spanclass="SimpleMath"D/> containingnice generatorsforvar="Arg>G and procedure towrite SLP for elements >G/var the nicegenerators code="">NeverApplicablecode if< class"">G</var> not; or< class">fail, the case no evidence found Arg">isagiant orevidencewas found, but the construction of >D<span> then calling becomesaleaf/>
<p>This method is used for recognizing permutation groups.</p>
<p>This method tries to determine whether the input group <var class="Arg">G</var
<p>A primitive group <span p method determine inputgroup class="Arg>is a giant that is, < class"">A_n<> or inin its action D nice for varclass="ArgG</var toan arbitrary of class"Arg>G from nice ;or"NeverApplicableG/var not ; orcode class="keyw>fail"G/ar>isa giant, evidencewas ,but theconstruction ofD
<p>A large primitive group <span class="SimpleMath">H</span> of the above type which does have fixed points is handled as follows: if the
<p>If <var class="Arg">G</var> is imprimitive then the output is <code class="keyw">NeverApplicable</code>. If <var class="Arg">Gjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
<p>This method is used for recognizing permutation groups.</p>
<p>If a permutation group moves only small points
<p>If the input group moves a large point (currently, this means a point larger than 10), then this method returns <code class="keyw">NeverApplicable</code>.<
<h5>6.2-6 <span class="Heading"><code class="code">NonTransitiveThis tries whether inputgroup <var="Arg"></ar afixed-point-free large-base group neither asymmetric an group its
<> groupspan="SimpleMath"><s> acting on <spanclassSimpleMath></> points called<>large</> thereexist class"SimpleMath">n<spanspan="SimpleMath>k/>,and< =SimpleMath"/> < class""> nchoose^<span> upto a permutational span ="">H/> is subgroup the wreathproduct< classSimpleMath">S_n ≀ S_r andanovergroup < =SimpleMath>() r/>< ="></spanspan=""A_n> actonthespan class=""><spansubsets =""{,. }/>.This recognisesfixed-point-free primitive withspan=""> k >1/>and< classSimpleMath2</pan./>
<p>This method is used for recognizing permutation groups.</p>
<p>If a permutation group <var class="Arg">G</var> acts nontransitively then this method computes a homomorphism to the action of <var class="Arg">G</var> on the orbit of the largest moved point
<p>This method is used for recognizing permutation <pIf permutationgroup class="Arg">G/var> acts then this computes a homomorphismto theaction <var="Arg">G</ar on the of the moved point Ifvar class"rg>G/> is transitive thenthe method returns keyw"NeverApplicable/code>.
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This method is used for recognizing permutation groups.</p>
<p>This methoddefineshomomorphism ofa permutation group <var class="rg">G/vartothe onthe points <var=ArgG<varifvar class"">G</var> has fixedpoints and eitherknown be primitive the of fixedpoints points exceeds certain threshold If <var="Arg"></var> has points is not primitive then itreturns <code classkeywNotEnoughInformation</code> so itmay called again atalater time allother, it <codeclass"keyw>everApplicable/code>./>
<p>In the current setup, the homomorphism is defined if the number <span class="SimpleMath">n</span> of moved points isjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The fact that this method returnsp>his methodi used forrecognizing groups.</java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<p>This method is successful if and only if all generators of a matrix group <var class="Arg">G</var> are diagonal matricesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Thisp>Note that this method is implemented in a way such that it also be asa forprojective group< class"">G<>. that the node the class=code!projective>component to ="keyw"true>andthis is passed to and.</p>
<p>This method defines a homomorphism from a matrix group <var class="Arg">G</var> into the projective group <var class="java.lang.StringIndexOutOfBoundsException: Range [0, 124) out of bounds for length 0
<> method is usedforrecognizing matrix groups.</p>
<p>fa stabilizer is , the node is knowledge thisknown chain, and image node told to homomorphism from the for permutationgroups a stabilizer of a parent is already this isused forfor theofstabilizerchain of this. Thisstabilizer is thenusedin same way asabove.</>
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Note that this method is implemented in a way such that it can also be used as a method for a projective group <var class="Arg">G</var>. In that case the recognition node has the <code class="code">!.projective</code> component bound to
<h5>6.3-10 <span class="Heading"><code classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This method is unused!</p>
<p>TODO</p>
<p>
<h4>6.4 <span class="Heading">Methods for projective groups</span></h4>
<p>The following table gives an overview over the installed methods and their rank (higher
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Projective group find homomorphism methods</caption>
<tr> ""3td
<td class="tdleft"><code class="code">TrivialGroup</h5 class=""<code/>
<td class="tdleft">go through generators and compare to the identity</td>
<td class="tdleft"><a href="chap6.html#X8489BECB78664847"><span<p>Thismethod isused for recognizing matrix andprojectivegroups.</p>
</tr>
<tr>
<td class="tdleft">1300</td>
<td class="tdleft"><code class="code"><p>Notethat method is inaway that it also used a method a groupvar="Arg"G/>. Inthat therecognition has <codeclass"">.projective/>component bound tocodeclass">andthisinformation passed toimage ./>
<td class="tdleft">find homomorphism to non-zero scalars mod d-th powers</td>
<tdclass"dleft">a hrefchap6html#5086487">6.4-20/span
</tr>
<tr>
<td class="tdleft">1200</td>
<td class="tdleft"><code class="code">ReducibleIso</code></td>
<td class="tdleft">use the MeatAxe to find invariant subspaces</td>
<td class="tdleft"><a href="chap6.html#X811FC944788D9839"><span class="RefLink">6.3-9</span></a></td>
</tr>
<tr>h46.4 <span class="Heading"Methodsfor groupsspan/h4
<td
<td class="tdleft"><code class="code">NotAbsolutelyIrred</code></td>
<tdclasstdleftwrite a biggerfield smaller degree/d>
<td class="tdleft"><a href="chap6.html#X83E6AD6583DB7080"><span class="RefLink">6.4-19</span>
</tr>
<tr>
<td class="tdleft">1050</td>
<td class="tdleft"><code class="code">ClassicalNatural</code></td>
<td<td class="tdleft">3000</td
< class">>6.4-9</></a></>
</tr>
<tr>
< class"10
<td class="tdleft"><code class<tr
<tdclasstdleft>rite a smallerfield same degree</td>
<td class="tdleft"><a href="chap6
900/td
<tdclasstdleft<code classcode"C3C5/code>
<td class="tdleft">compute a normal subgroup of derived and resolve C3 and/>
<td class="tdleft"><a href="chap6.html#X799F4A7A8239D70F"><span class="RefLink">6.4-7</span></a></td>
</tr>
<tr>
<td class= classu the find subspaces>
<td class="tdleft"><code <d class="tdleft"<a href="chap6.#X811FC944788D9839"<span class="">639/></a>/td
<td class="tdleft">find either an (imprimitive) action or a symplectic one</td>
<td class="tdleft"><a href="chap6.html#X7E484B8F83849B49"><span class="RefLink">6.4-8</span></a></td>
</tr>
<tr>
<td class="tdleft">840/td
<td class="tdleft"><code class="code">D247</code></td>
<td class="tdleft">play games to find a normal subgroup</td>
<td class="tdleft"><a href="chap6.html#X81E155F98671824A"><span class="RefLink">6.4-11</span></a></td>
</tr>
<tr>
<td class="tdleft">810</td>
<td class="tdleft"><code class="code">AltSymBBByDegree</code></td>
<td class="tdleft">try BB recognition for dim+1td="">check it a classical group its </>
<td class="tdleft"><a href="chap6.html#X87A53FB6813E2E3C"><span class="RefLink">6.4-1</span></a></td>
</tr>
<tr>
<td class="tdleft">800</td>
<td class="tdleft"><code class="code">TensorDecomposable</code></td>
<td class="tdleft">find a tensor decomposition</td>
<tdtr>
</tr>
<>
<td class="tdleft">700</td>
<td class="tdleft"><code class="code">FindElmOfEvenNormal</code></td>
<td class="tdleft">find D2, D4 or D7 by finding an element of an even normal subgroup</td>
<td class="tdleft"><a href="chap6.html#X7ED3241F84A98292"><span class="RefLink">6.4-13</span></a></td>
</r>
<tr>
< class">0
<td class="tdleft"><code class="code">LowIndex</code></td>
<td class="tdleft">find an (imprimitive) action on subspaces</td>
<tdclass"tdleft"><a href="chap6html#X810B7F36818D626B"<class">6.-7/>
</tr>
<tr>
<td class="tdleft">580</td>
<td class"code>/</>
<td class="tdleft">generate maximal orders</td>
<td class="tdleft"><a href="chap6.html#X82B704EF80AE2FBB"><span class="RefLink">6.4-18</span></a></td>
</tr>
<tr>
<tdclass="">55</d>
<td class="tdleft"><code class="code">ComputeSimpleSocle</code></td>
<td class="tdleft">compute</tr
<td class="tdleft"><a href="java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 27
</tr>
<tr>
t><>
<td class="tdleft"><code class="code">ThreeLargeElOrders</code></td>
<td class="tdleft">recognise Lie type groups and get its classtdleft8<td
<td class="tdleft"><a href="chap6.html#X7A82C26E78FEF78F"><span class="RefLink">6.4-26</span></a></td>
</tr>
<r
<td class="tdleft">400</td>
<td class="tdleft"><code class="code">LieTypeNonConstr</code></td>
<td class="tdleft">do non-constructive recognition of Lie type groups</td>
<td class="tdleft"><a href="chap6.html#X865D3A1B7B6BB49B"><span class="RefLink">6.4-16</span></a>< classtdleft D2D4 D7 findingan element an normal</td
</tr>
<tr>
<td class="tdleft">100</td>td="tdleft">600/>
<td class="tdleft"><code class="code">StabilizerChainProj</code></td>
< classtdleftlast: compute stabilizer chain (rojectively)/>
<td class="tdleft"><a href= =RefLink4-7/pan</>
</tr>
</table<br />
</divtr
<h5>6.4-2 <span class="Heading"><code class="code">BiggerScalarsOnly</code></span></h5td=""><codejava.lang.StringIndexOutOfBoundsException: Range [30, 26) out of bounds for length 68
<p>This method is only called when hinted from above. In this method
<p>This method does the balanced tree approach mapping to subsets of the diagonal blocks and finally using projective recognition to recognise single diagonal block groups
<p>This method is used for recognizing projective groups.</p>
<p>This method is designed for the handling of the Aschbacher class C6 (normaliser of an extraspecialjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<This computes non-abeliansimple and italong additional if iscalled an almost simple group. Once the non-abelian simple socle is computed the function does not need to be called again for this node and therefore returns <code class="keyw">NeverApplicable</code>.</p>
<p>This method is used for recognizing projective groups.</p>
<p>Recognise quasi-simple group of Lie type when characteristic is given. Based on <a href="chapBib.html#biBBKPS02">[BKPS02]</a> and <a href="chapBib.html#biBAB01">[AB01
<p>This method is used for recognizing projective groups.</p>
<p>This method is designed for the handling of the Aschbacher class C2 (stabiliser of a decomposition of the java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 0
<p>This method is used for recognizing projective groups.</p>
<p>If an irreducible projective group <var class="Arg">G</var> acts absolutely irreducibly then this method returns <code class="keyw">NeverApplicable</#biBAB01">AB01]/><p
<p>This method is used for recognizing projective groups.</p>
<p>TODO/FIXME: it is unclear if the following description actually belongs to this method, so be cautious!</p>
<>his methodcurrently to onetensor factor by powering up commutators of random elements toelements .This toworkwellprovided that thetwo arenot qlinked/q> such that thereexistenough elementsthatact with different factors<pjava.lang.StringIndexOutOfBoundsException: Index 325 out of bounds for length 325
<p>This method and its description needs some improvement.</p>
<p><a id="X7A82C26E78FEF78F" name="X7A82C26E78FEF78F"></a><pThismethodisused for recognizing groups</java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
<p>This method is used for recognizing projective groupsp><a id="" name""<a<p>
<p>In the case when the input group <var class="Arg">G</var><span class="SimpleMath">≤ PGL(d,p^e)</
<p>This method assumes that its input is neither alternating nor sporadic and that <code class="func">ComputeSimpleSocle</code> (<a href="chap6.html#X7EC8D0FD7DC6B213"><span class="RefLink">6.4-10</span></a>) has already been called.</p>
<p>This recognition method is based on the paper <a href="chapBib.html#biBKS09">[KS09]</a>.</p>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Topjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
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