<div java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<div class="chlinkprevnexttop"> <a meta name="generator" content="GAPDoc2HTML=sjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 59
<p id="script type="textjava.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 57
<p><a id=
<java.lang.StringIndexOutOfBoundsException: Range [21, 4) out of bounds for length 158
</ivjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<h3>2 <span class
<p>Before explaining the theory behind the <strong class="pkg">java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">K := GF( 2 );</span<p><a="X7AAB1F18838B1DAB"name"AAB1F18838B1DAB"//
GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralGroup( 16 );</span>
<h3>2span class="Heading"A calculationwithclass=pkg></strong><span</h3>
<span
<>Before explaining theory the <trong class=pkg"LAGUNA/strong> package present a the reader what <strong class=""LAGUNA</strong> is able to compute. We will carry out some calculations in the group algebra of the dihedral group of order 16 over the field of two elements. First we create this modular group
</pre></div>
<>The groupalgebra class="ode"KG/code has somepropertiesand that direct ofits definition These can bechecked veryquickly./p>
<iv class>pre
<span class="lt;-with2, &;
true
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFModularGroupAlgebra( KG );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingGroup
& 4 &java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
<span KG );</span>
GF(2)
</pre></div>
<p>Since <java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 4
<span class="GAPprompt">gap&span class="GAPprompt">gap>><spanclass="APinput">eftActingDomain )</span>
GF(2)
<spanp> <code class="ode"KG/java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 201
GF2)
</pre></div>
<p>Let us construct a java.lang.StringIndexOutOfBoundsException: Range [0, 29) out of bounds for length 0
<div class="example"><pre>
<span lass="APpromptg;/span>< class="APinput>UnderlyingField KG)<spanjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
[f1 f2 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
<span class
[((2)0)f1, (Z2)0)f2 ]
</pre></div>
<p>Now we construct java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 10
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(Z(2)^0)*f1
(Z(2)^0)*f2
<span class="GAPprompt">gap></span> <span class="GAPinput">e := One(
(Z(2)^0)*<identity> of ...
<pan class="GAPprompt"gap&t/pan> < class=GAPinput"x :=(e + )*(e + b ); /span>
((2)0)<dentity&t of ..(Z()^)f1Z(2)^)f2(()^)f1*
</pre></java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
<
[ <
<=g>java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 96
[ Z(2)
<span<iv class=example"<re> 4
<span class="GAPprompt">gap></span> <span class="GAPinput">TraceOfMagmaRingElement[<dentity&t;of.. f1 java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 42
Z(2)^0
</pre></div>Z(),(2^, Z2)0,Z(^0 ]
<p calculatethe augmentationof <ode class"ode>x/code>, which is defined as the sum of its coefficients.</p>
<div class="xample">pre>
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 0*Z(2)
<span class="GAPprompt">gap></span> <span
false
<spanfalse
(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span
p>Sin theaugmentationof<codeclass"ode> is zero,">1+x/>is Thisagainveryeasytocheck<p
</pre></div>
<p><strong class="pkg">LAGUNA</strong> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="example"><pre>
<spanclass="APprompt"gap&;/pan><spanclass=GAPinput">sUnit(KG y )<span>
(Z(2)^0)*f1+(Z(2)^0)*f2+(true 0)*f1*f3+(Z(2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Z2)0)*
<span class="GAPprompt">gap&
(()0*lt;identity> of ...
</pre></div>
<p>We may also want to2^*(()0)*((2^)+(20*1*+Z(^
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Involution()0)f1*2*
(Z(2)^0)*f1+(Z(2)^0)*()0*&; .
< class"APprompt"gap&t<span<span class=GAPinput> Involution )/span
false
< class"APprompt">gap></span> <span class="GAPinput">IsSymmetric( y );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">y * Involution( y );</span>
(Z(2)0)lt;dentity&;of .+Z2)0)f2+Z(2)0)f2**f4
<span class="GAPprompt">gap&spanclass="GAPprompt">gap></span> <spanclass="GAPinput">y = Involution( y );</java.lang.StringIndexOutOfBoundsException: Range [0, 88) out of bounds for length 5
false
</pre></div>
<p>Now we calculateZ)^)<identityg;of..+Z(2^)f2(2^)*2*f3*f4
<div""<java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap></span
&java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 78
); span=>>span<class>java.lang.StringIndexOutOfBoundsException: Range [79, 78) out of bounds for length 118
java.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for length 4
</>/div
<>Itis -knownthat the augmentation ideal <codeclass""KG/>is nilpotent ideal. UsingJennings'theory on dimension subgroups,we obtain its nilpotency immediatecalculation of its powers.This implemented in strong class="pkg">LAGUNA</strong>.</p>
<div
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0 9
</pre></div>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 1
<span class="APprompt">gap></span> <span class="GAPinput">s[2];</span>
<algebra of java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<panclass="APprompt>&;<span> <span class="GAPinput">List(s,Dimension);</span>
[< class=GAPprompt>><span><span class="APinput"s];/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(s);</span> 9
</pre></div>
<>Wesee that thelength of this list is exactlythe index augmentation ideal of<codeclass="ode>KG</code>.<p>
<p>Now java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
< class="xample"<re
<span class=pNow ' java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 413
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 49
</pre></div>
< ourjava.lang.StringIndexOutOfBoundsException: Range [38, 35) out of bounds for length 140
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"><group of size32768with15 generators&t
<pc java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
</pre></div>
<>strong class=p></strong has manyefficient andpractical algorithmsfor groups given bya power- presentation.In to these algorithmsto carry outcomputationin thenormalised unitgroup,we tosetup between the outputsof <odeclass=code">NormalizedUnitGroup<code> and <code class="code">PcNormalizedUnitGroup</code>.</p>
<div class="example"><><strongclass=pkg></trong hasmanyefficient practicalalgorithmsforgroups given byapowercommutator presentation In order to these to carryout computation in the normalised unit group,we toset up isomorphismsbetweentheoutputsof code class="ode"NormalizedUnitGroup</ode>and<code class="ode>PcNormalizedUnitGroup</code>.</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := NaturalBijectionToPcNormalizedUnitGroup( KG );</span>
MappingByFunction( 32768 with<span class=GAPprompt>&t;/pan span class="GAPinput">t := NaturalBijectionToPcNormalizedUnitGroup( KG );</span>
=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 82
true
<span class="GAPprompt">gap> 32768gg;(xjava.lang.StringIndexOutOfBoundsException: Range [54, 55) out of bounds for length 54
ltidentity&t .., f1 f2,f4 f8, f1* f1*f4,f1*f8,f2*,f2*,
f4*java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<div class="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 12
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</pre></div>
<p>For example, wespan class=GAPprompt>><span><panclass="APinput"Imagef ;span>
=>>
<span java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
< "">/ span =G"(cc )<span> 848
<span class="GAPprompt"span =""g;/ class"APinput>(cc )<span>
f1java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
(Z(2)^0)*<identity*f3* 0**f3(2^**f2((2*1*f4
</pre></div>
<p>Having Z2)0)ltidentitygt of ..((2^0)*(2)^)*1*3+Z()^0)**f4(()java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
<div class="example"/pre>/iv>
<class="APprompt"gap&t;<span<class=GAPinput( W )<span>
[ <pc java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
Group([ f4*f8java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
f10*f11, f12*f13,f13*,f14*f15],
Group([ f8, f9*f15, f10*f11, f12*f15, f13* <pc group of size 32768 with 15 generators
java.lang.StringIndexOutOfBoundsException: Range [31, 7) out of bounds for length 76
</pre></div>
<p>Let's now compute, for instance, a minimal system of generators of the centre of the normalised java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 61
<div class="<pre></iv>
<span class="APprompt"gap&</span><span lassGAPinput> := Centre( W);</span>
<span class
[ f8*13**f15, 13f14*,f8**f14*, f15,f4*f6*8f13]
<span class="span class="GAPpromptgap&;/span><span class="GAPinput">C := Centre( W );;</span>
s =G>&java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 100
*f3(2)*ff2
(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*span class="GAPprompt">gap></span=GAPinput>( g-gt;gf );/
Z)^0)f1*f4(()0)f1*f2*f3*, ((2)^)&;dentity&t of..+Z()^ 0)*f1+(Z(2)^0)*f3+(Z 0)f1*f2f3(Z()^0**f2*4(()0)f1*f2*3f4 0*f3*f4+(Z(2)0***f3+Z2)^0)f1**f4+((2)0*f1f3*+((2) 0)* Z(2)^*1f2*+(()^0)f1*f3* ((20)&;dentity&;of..(()^ 0)*f10*f1+Z)^0)f3+(Z(2)0*f1*f2+((2)0**3+Z()^)f1f4+(()^
Z)^0)f3*f4+(()0)*1*f2*f3+Z2)^)f1*f2*f4+(Z)^)*1*3f4Z() 0*f2*f3*f4+((2)^0*f1*f2*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(
Z)^)f4(Z()0)f1*+Z2^0*f1*f4+Z2)0)f3f4(Z()java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66 0)*f1**f4+Z(2)0*f2**4 ]
</pre></div>
<>We finishourexample by somepropertiesof theLiealgebra associatedwith <codeclass=""KG</>.This needsno furtherexplanation<p
<div class="example"><pre>
<span class="
#I package:ConstructingLiealgebra ...
<Lie algebra of dimension16 over GF()gt;
<span class="GAPpromptjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#I java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
&;
<span class="GAPprompt">gap></span> <span class="GAPinput">LC := LieCentre(
&t algebra of dimension 7 over GF(2)>
<span class="GAPprompt">gap&span class="GAPprompt">gap></span> <span class="GAPinput">DILAGUNA . 5
<span class="GAPprompt"gapgt<span><span class"APinput"LC: LieCentreL)/span> 5
<span class="<pan class="GAPprompt>gap&;/> < class="APinput"LieLowerNilpotencyIndexKG)<span>
false
<span classjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
#I LAGUNA package: Checking Lie solvability ...
true
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 5
false
<span class="GAPprompt">gap></span> <span class="#I LAGUNA package: Checking Lie solvability ...
true
</pre></div>
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