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<div class="ChapSects"><a href="chap3_mj.html#X81CAD2F27B2066C4">3 <span class="Heading">An example application</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7DAC33E37B977087">3.1 <span class="Heading">Presentation for rational matrix groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X79CF643081B3FB26">3.2 <span class="Heading">Modules series</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7BA34CD28059D6CD">3.3 <span class="Heading">Triangularizable subgroups</span></a>
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<h3>3 <span class="Heading">An example application</span></h3>

<p>In this section we outline three example computations with functions from the previous chapter.</p>

<p><a id="X7DAC33E37B977087" name="X7DAC33E37B977087"></a></p>

<h4>3.1 <span class="Heading">Presentation for rational matrix groups</span></h4>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats :=</span>
[ [ [ 1, 0, -1/2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ 1, 1/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ],
  [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ 1, -1/2, -3, 7/6 ], [ 0, 1, -1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ -1, 3, 3, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ] ];

<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( mats );</span>
<matrix group with 5 generators>

# calculate an isomorphism from G to a pcp-group
<span class="GAPprompt">gap></span> <span class="GAPinput">nat := IsomorphismPcpGroup( G );;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">H := Image( nat );</span>
Pcp-group with orders [ 2, 2, 3, 5, 5, 5, 0, 0, 0 ]

<span class="GAPprompt">gap></span> <span class="GAPinput">h := GeneratorsOfGroup( H );</span>
[ g1, g2, g3, g4, g5, g6, g7, g8, g9]

<span class="GAPprompt">gap></span> <span class="GAPinput">mats2 := List( h, x -> PreImage( nat, x ) );;</span>

# take a random element of G
<span class="GAPprompt">gap></span> <span class="GAPinput">exp :=  [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := MappedVector( exp, mats2 );</span>
[ [ -1, 17/2, -1, 233/6 ],
  [ 0, 1, 0, -2 ],
  [ 0, 1, -1, 2 ],
  [ 0, 0, 0, 1 ] ]

# map g into the image of nat
<span class="GAPprompt">gap></span> <span class="GAPinput">i := ImageElm( nat, g );</span>
g1*g2*g3*g4*g9

# exponent vector
<span class="GAPprompt">gap></span> <span class="GAPinput">Exponents( i );</span>
[ 1, 1, 1, 1, 0, 0, 0, 0, 1 ]

# compare the preimage with g
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( nat, i );</span>
[ [ -1, 17/2, -1, 233/6 ],
  [ 0, 1, 0, -2 ],
  [ 0, 1, -1, 2 ],
  [ 0, 0, 0, 1 ] ]

<span class="GAPprompt">gap></span> <span class="GAPinput">last = g;</span>
true

</pre></div>

<p><a id="X79CF643081B3FB26" name="X79CF643081B3FB26"></a></p>

<h4>3.2 <span class="Heading">Modules series</span></h4>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens :=</span>
[ [ [ 1746/1405, 524/7025, 418/1405, -77/2810 ],
    [ 815/843, 899/843, -1675/843, 415/281 ],
    [ -3358/4215, -3512/21075, 4631/4215, -629/1405 ],
    [ 258/1405, 792/7025, 1404/1405, 832/1405 ] ],
  [ [ -2389/2810, 3664/21075, 8942/4215, -35851/16860 ],
    [ 395/281, 2498/2529, -5105/5058, 3260/2529 ],
    [ 3539/2810, -13832/63225, -12001/12645, 87053/50580 ],
    [ 5359/1405, -3128/21075, -13984/4215, 40561/8430 ] ] ];

<span class="GAPprompt">gap></span> <span class="GAPinput">H := Group( gens );</span>
<matrix group with 2 generators>

<span class="GAPprompt">gap></span> <span class="GAPinput">RadicalSeriesSolvableMatGroup( H );</span>
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ 1, 0, 0, 79/138 ], [ 0, 1, 0, -275/828 ], [ 0, 0, 1, -197/414 ] ],
  [ [ 1, 0, -3, 2 ], [ 0, 1, 55/4, -55/8 ] ],
  [ [ 1, 4/15, 2/3, 1/6 ] ],
  [  ] ]
</pre></div>

<p><a id="X7BA34CD28059D6CD" name="X7BA34CD28059D6CD"></a></p>

<h4>3.3 <span class="Heading">Triangularizable subgroups</span></h4>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := PolExamples(3);</span>
<matrix group with 2 generators>

<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( G );</span>
[ [ [ 73/10, -35/2, 42/5, 63/2 ],
    [ 27/20, -11/4, 9/5, 27/4 ],
    [ -3/5, 1, -4/5, -9 ],
    [ -11/20, 7/4, -2/5, 1/4 ] ],
  [ [ -42/5, 423/10, 27/5, 479/10 ],
    [ -23/10, 227/20, 13/10, 231/20 ],
    [ 14/5, -63/5, -4/5, -79/5 ],
    [ -1/10, 9/20, 1/10, 37/20 ] ] ]

<span class="GAPprompt">gap></span> <span class="GAPinput">subgroups := SubgroupsUnipotentByAbelianByFinite( G );</span>
rec( T := <matrix group with 2 generators>,
  U := <matrix group with 4 generators> )

<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( subgroups.T );</span>
[ [ [ 73/10, -35/2, 42/5, 63/2 ],
    [ 27/20, -11/4, 9/5, 27/4 ],
    [ -3/5, 1, -4/5, -9 ],
    [ -11/20, 7/4, -2/5, 1/4 ] ],
  [ [ -42/5, 423/10, 27/5, 479/10 ],
    [ -23/10, 227/20, 13/10, 231/20 ],
    [ 14/5, -63/5, -4/5, -79/5 ],
    [ -1/10, 9/20, 1/10, 37/20 ] ] ]

# so G is triangularizable!
</pre></div>

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