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<div class="ChapSects"><a href="chap2_mj.html#X829BA50B82FEC109">2 <span class="Heading">Methods for matrix groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X826C51B3825A2789">2.1 <span class="Heading">Polycyclic presentations of matrix groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7A1BC4437FD92201">2.1-1 PcpGroupByMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X8771540F7A235763">2.1-2 IsomorphismPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X85ADB89B7C8DD7D0">2.1-3 ImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X809C78D5877D31DF">2.1-4 IsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7EE01C207C214C1F">2.1-5 IsTriangularizableMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7D7456077D3D1B86">2.1-6 IsPolycyclicGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X80D1E9E07DB87F97">2.2 <span class="Heading">Module series</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X84472FDC863322BD">2.2-1 RadicalSeriesSolvableMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X8524F992828B6A71">2.2-2 HomogeneousSeriesAbelianMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X87D9F67C7CBB1499">2.2-3 HomogeneousSeriesTriangularizableMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X86FB6E9B801A37D4">2.2-4 CompositionSeriesAbelianMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X78DE110C7E2A493C">2.2-5 CompositionSeriesTriangularizableMatGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7BA181CA81D785BB">2.3 <span class="Heading">Subgroups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X79273B8581D15356">2.3-1 SubgroupsUnipotentByAbelianByFinite</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7A489A5D79DA9E5C">2.4 <span class="Heading">Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7C7C3EFA7E49F932">2.4-1 PolExamples</a></span>
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<h3>2 <span class="Heading">Methods for matrix groups</span></h3>
<p><a id="X826C51B3825A2789" name="X826C51B3825A2789"></a></p>
<h4>2.1 <span class="Heading">Polycyclic presentations of matrix groups</span></h4>
<p>Groups defined by polycyclic presentations are called PcpGroups in <strong class="pkg">GAP</strong>. We refer to the Polycyclic manual <a href="chapBib_mj.html#biBPolycyclic">[EN00]</a> for further background.</p>
<p>Suppose that a collection <span class="SimpleMath">\(X\)</span> of matrices of <span class="SimpleMath">\(GL(d,R)\)</span> is given, where the ring <span class="SimpleMath">\(R\)</span> is either <span class="SimpleMath">\(ℚ,ℤ\)</span> or a finite field. Let <span class="SimpleMath">\(G= \langle X \rangle\)</span>. If the group <span class="SimpleMath">\(G\)</span> is polycyclic, then the following functions determine a PcpGroup isomorphic to <span class="SimpleMath">\(G\)</span>.</p>
<p><a id="X7A1BC4437FD92201" name="X7A1BC4437FD92201"></a></p>
<h5>2.1-1 PcpGroupByMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PcpGroupByMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">G</var> is a subgroup of <span class="SimpleMath">\(GL(d,R)\)</span> where <span class="SimpleMath">\(R=ℚ,ℤ \)</span> or <span class="SimpleMath">\(\mathbb{F}_q\)</span>. If <var class="Arg">G</var> is polycyclic, then this function determines a PcpGroup isomorphic to <var class="Arg">G</var>. If <var class="Arg">G</var> is not polycyclic, then this function returns <code class="code">fail</code>.</p>
<p><a id="X8771540F7A235763" name="X8771540F7A235763"></a></p>
<h5>2.1-2 IsomorphismPcpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsomorphismPcpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><var class="Arg">G</var> is a subgroup of <span class="SimpleMath">\(GL(d,R)\)</span> where <span class="SimpleMath">\(R=ℚ,ℤ \)</span> or <span class="SimpleMath">\(\mathbb{F}_q\)</span>. If <var class="Arg">G</var> is polycyclic, then this function determines an isomorphism onto a PcpGroup. If <var class="Arg">G</var> is not polycyclic, then this function returns <code class="code">fail</code>.</p>
<p>Note that the method <code class="code">IsomorphismPcpGroup</code>, installed in this package, cannot be applied directly to a group given by the function <code class="code">AlmostCrystallographicGroup</code>. Please use <code class="code">POL_AlmostCrystallographicGroup</code> (with the same parameters as <code class="code">AlmostCrystallographicGroup</code>) instead.</p>
<p><a id="X85ADB89B7C8DD7D0" name="X85ADB89B7C8DD7D0"></a></p>
<h5>2.1-3 ImagesRepresentative</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImagesRepresentative</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImageElm</code>( <var class="Arg">map</var>, <var class="Arg">elm</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImagesSet</code>( <var class="Arg">map</var>, <var class="Arg">elms</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Here <var class="Arg">map</var> is an isomorphism from a polycyclic matrix group <var class="Arg">G</var> onto a PcpGroup <var class="Arg">H</var> calculated by <code class="func">IsomorphismPcpGroup</code> (<a href="chap2_mj.html#X8771540F7A235763"><span class="RefLink">2.1-2</span></a>). These methods can be used to compute with such an isomorphism. If the input <var class="Arg">elm</var> is an element of <var class="Arg">G</var>, then the function <code class="code">ImageElm</code> can be used to compute the image of <var class="Arg">elm</var> under <var class="Arg">map</var>. If <var class="Arg">elm</var> is not contained in <var class="Arg">G</var> then the function <code class="code">ImageElm</code> returns <code class="code">fail</code>. The input <var class="Arg">pcpelm</var> is an element of <var class="Arg">H</var>.</p>
<p><a id="X809C78D5877D31DF" name="X809C78D5877D31DF"></a></p>
<h5>2.1-4 IsSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSolvableGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><var class="Arg">G</var> is a subgroup of <span class="SimpleMath">\(GL(d,R)\)</span> where <span class="SimpleMath">\(R=ℚ,ℤ \)</span> or <span class="SimpleMath">\(\mathbb{F}_q\)</span>. This function tests if <var class="Arg">G</var> is solvable and returns <code class="code">true</code> or <code class="code">false</code>.</p>
<p><a id="X7EE01C207C214C1F" name="X7EE01C207C214C1F"></a></p>
<h5>2.1-5 IsTriangularizableMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsTriangularizableMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">G</var> is a subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>. This function tests if <var class="Arg">G</var> is triangularizable (possibly over a finite field extension) and returns <code class="code">true</code> or <code class="code">false</code>.</p>
<p><a id="X7D7456077D3D1B86" name="X7D7456077D3D1B86"></a></p>
<h5>2.1-6 IsPolycyclicGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPolycyclicGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><var class="Arg">G</var> is a subgroup of <span class="SimpleMath">\(GL(d,R)\)</span> where <span class="SimpleMath">\(R=ℚ,ℤ \)</span> or <span class="SimpleMath">\(\mathbb{F}_q\)</span>. This function tests if <var class="Arg">G</var> is polycyclic and returns <code class="code">true</code> or <code class="code">false</code>.</p>
<p><a id="X80D1E9E07DB87F97" name="X80D1E9E07DB87F97"></a></p>
<h4>2.2 <span class="Heading">Module series</span></h4>
<p>Let <span class="SimpleMath">\(G\)</span> be a finitely generated solvable subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>. The vector space <span class="SimpleMath">\(ℚ^d\)</span> is a module for the algebra <span class="SimpleMath">\(ℚ[G]\)</span>. The following functions provide the possibility to compute certain module series of <span class="SimpleMath">\(ℚ^d\)</span>. Recall that the radical <span class="SimpleMath">\(Rad_G(ℚ^d)\)</span> is defined to be the intersection of maximal <span class="SimpleMath">\(ℚ[G]\)</span>-submodules of <span class="SimpleMath">\(ℚ^d\)</span>. Also recall that the radical series</p>
<p class="center">\[
0=R_n < R_{n-1} < \dots < R_1 < R_0=ℚ^d
\]</p>
<p>is defined by <span class="SimpleMath">\(R_{i+1}:= Rad_G(R_i)\)</span>.</p>
<p><a id="X84472FDC863322BD" name="X84472FDC863322BD"></a></p>
<h5>2.2-1 RadicalSeriesSolvableMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RadicalSeriesSolvableMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function returns a radical series for the <span class="SimpleMath">\(ℚ[G]\)</span>-module <span class="SimpleMath">\(ℚ^d\)</span>, where <var class="Arg">G</var> is a solvable subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>.</p>
<p>A radical series of <span class="SimpleMath">\(ℚ^d\)</span> can be refined to a homogeneous series.</p>
<p><a id="X8524F992828B6A71" name="X8524F992828B6A71"></a></p>
<h5>2.2-2 HomogeneousSeriesAbelianMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HomogeneousSeriesAbelianMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A module is said to be homogeneous if it is the direct sum of pairwise irreducible isomorphic submodules. A homogeneous series of a module is a submodule series such that the factors are homogeneous. This function returns a homogeneous series for the <span class="SimpleMath">\(ℚ[G]\)</span>-module <span class="SimpleMath">\(ℚ^d\)</span>, where <var class="Arg">G</var> is an abelian subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>.</p>
<p><a id="X87D9F67C7CBB1499" name="X87D9F67C7CBB1499"></a></p>
<h5>2.2-3 HomogeneousSeriesTriangularizableMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HomogeneousSeriesTriangularizableMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A module is said to be homogeneous if it is the direct sum of pairwise irreducible isomorphic submodules. A homogeneous series of a module is a submodule series such that the factors are homogeneous. This function returns a homogeneous series for the <span class="SimpleMath">\(ℚ[G]\)</span>-module <span class="SimpleMath">\(ℚ^d\)</span>, where <var class="Arg">G</var> is a triangularizable subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>.</p>
<p>A homogeneous series can be refined to a composition series.</p>
<p><a id="X86FB6E9B801A37D4" name="X86FB6E9B801A37D4"></a></p>
<h5>2.2-4 CompositionSeriesAbelianMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CompositionSeriesAbelianMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A composition series of a module is a submodule series such that the factors are irreducible. This function returns a composition series for the <span class="SimpleMath">\(ℚ[G]\)</span>-module <span class="SimpleMath">\(ℚ^d\)</span>, where <var class="Arg">G</var> is an abelian subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>.</p>
<p><a id="X78DE110C7E2A493C" name="X78DE110C7E2A493C"></a></p>
<h5>2.2-5 CompositionSeriesTriangularizableMatGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CompositionSeriesTriangularizableMatGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A composition series of a module is a submodule series such that the factors are irreducible. This function returns a composition series for the <span class="SimpleMath">\(ℚ[G]\)</span>-module <span class="SimpleMath">\(ℚ^d\)</span>, where <var class="Arg">G</var> is a triangularizable subgroup of <span class="SimpleMath">\(GL(d,ℚ)\)</span>.</p>
<p><a id="X7BA181CA81D785BB" name="X7BA181CA81D785BB"></a></p>
<h4>2.3 <span class="Heading">Subgroups</span></h4>
<p><a id="X79273B8581D15356" name="X79273B8581D15356"></a></p>
<h5>2.3-1 SubgroupsUnipotentByAbelianByFinite</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SubgroupsUnipotentByAbelianByFinite</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">G</var> is a subgroup of <span class="SimpleMath">\(GL(d,R)\)</span> where <span class="SimpleMath">\(R=ℚ\)</span> or <span class="SimpleMath">\(ℤ\)</span>. If <var class="Arg">G</var> is polycyclic, then this function returns a record containing two normal subgroups <span class="SimpleMath">\(T\)</span> and <span class="SimpleMath">\(U\)</span> of <span class="SimpleMath">\(G\)</span>. The group <span class="SimpleMath">\(T\)</span> is unipotent-by-abelian (and thus triangularizable) and of finite index in <var class="Arg">G</var>. The group <span class="SimpleMath">\(U\)</span> is unipotent and is such that <span class="SimpleMath">\(T/U\)</span> is abelian. If <var class="Arg">G</var> is not polycyclic, then the algorithm returns <code class="code">fail</code>.</p>
<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<h4>2.4 <span class="Heading">Examples</span></h4>
<p><a id="X7C7C3EFA7E49F932" name="X7C7C3EFA7E49F932"></a></p>
<h5>2.4-1 PolExamples</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PolExamples</code>( <var class="Arg">l</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns some examples for polycyclic rational matrix groups, where <var class="Arg">l</var> is an integer between 1 and 24. These can be used to test the functions in this package. Some of the properties of the examples are summarised in the following table.</p>
<div class="example"><pre>
PolExamples number generators subgroup of Hirsch length
1 3 GL(4,Z) 6
2 2 GL(5,Z) 6
3 2 GL(4,Q) 4
4 2 GL(5,Q) 6
5 9 GL(16,Z) 3
6 6 GL(4,Z) 3
7 6 GL(4,Z) 3
8 7 GL(4,Z) 3
9 5 GL(4,Q) 3
10 4 GL(4,Q) 3
11 5 GL(4,Q) 3
12 5 GL(4,Q) 3
13 5 GL(5,Q) 4
14 6 GL(5,Q) 4
15 6 GL(5,Q) 4
16 5 GL(5,Q) 4
17 5 GL(5,Q) 4
18 5 GL(5,Q) 4
19 5 GL(5,Q) 4
20 7 GL(16,Z) 3
21 5 GL(16,Q) 3
22 4 GL(16,Q) 3
23 5 GL(16,Q) 3
24 5 GL(16,Q) 3
</pre></div>
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