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<div class="ChapSects"><a href="chap8.html#X796AB9787E2A752C">8 <span class="Heading">Cohomology for pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X875758FA7C6F5CE1">8.1 <span class="Heading">Cohomology records</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7C97442C7B78806C">8.1-1 CRRecordByMats</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8646DFA1804D2A11">8.1-2 CRRecordBySubgroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X874759D582393441">8.2 <span class="Heading">Cohomology groups</span></a>
</span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X85EF170387D39D4A">8.2-1 OneCoboundariesCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X79B48D697A8A84C8">8.2-2 TwoCohomologyModCR</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X79610E9178BD0C54">8.3 <span class="Heading">Extended 1-cohomology</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7E87E3EA81C84621">8.3-1 OneCoboundariesEX</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8111D2087C16CC0C">8.3-2 OneCocyclesEX</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X84718DDE792FB212">8.3-3 OneCohomologyEX</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X853E51787A24AE00">8.4 <span class="Heading">Extensions and Complements</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7DA9162085058006">8.4-1 ComplementCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7F8984D386A813D6">8.4-2 ComplementsCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7FAB3EB0803197FA">8.4-3 ComplementClassesCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8759DC59799DD508">8.4-4 ComplementClassesEfaPcps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7B0EC76D81A056AB">8.4-5 ComplementClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X85F3B55C78CF840B">8.4-6 ExtensionCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X81DC85907E0948FD">8.4-7 ExtensionsCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7AE16E3687E14B24">8.4-8 ExtensionClassesCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7986997B78AD3292">8.4-9 SplitExtensionPcpGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X823771527DBD857D">8.5 <span class="Heading">Constructing pcp groups as extensions</span></a>
</span>
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<h3>8 <span class="Heading">Cohomology for pcp-groups</span></h3>
<p>The <strong class="pkg">GAP</strong> 4 package <strong class="pkg">Polycyclic</strong> provides methods to compute the first and second cohomology group for a pcp-group <span class="SimpleMath">U</span> and a finite dimensional <span class="SimpleMath">ℤ U</span> or <span class="SimpleMath">FU</span> module <span class="SimpleMath">A</span> where <span class="SimpleMath">F</span> is a finite field. The algorithm for determining the first cohomology group is outlined in <a href="chapBib.html#biBEic00">[Eic00]</a>.</p>
<p>As a preparation for the cohomology computation, we introduce the cohomology records. These records provide the technical setup for our cohomology computations.</p>
<p><a id="X875758FA7C6F5CE1" name="X875758FA7C6F5CE1"></a></p>
<h4>8.1 <span class="Heading">Cohomology records</span></h4>
<p>Cohomology records provide the necessary technical setup for the cohomology computations for polycyclic groups.</p>
<p><a id="X7C97442C7B78806C" name="X7C97442C7B78806C"></a></p>
<h5>8.1-1 CRRecordByMats</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CRRecordByMats</code>( <var class="Arg">U</var>, <var class="Arg">mats</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates an external module. Let <var class="Arg">U</var> be a pcp group which acts via the list of matrices <var class="Arg">mats</var> on a vector space of the form <span class="SimpleMath">ℤ^n</span> or <span class="SimpleMath">F_p^n</span>. Then this function creates a record which can be used as input for the cohomology computations.</p>
<p><a id="X8646DFA1804D2A11" name="X8646DFA1804D2A11"></a></p>
<h5>8.1-2 CRRecordBySubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CRRecordBySubgroup</code>( <var class="Arg">U</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CRRecordByPcp</code>( <var class="Arg">U</var>, <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates an internal module. Let <var class="Arg">U</var> be a pcp group and let <var class="Arg">A</var> be a normal elementary or free abelian normal subgroup of <var class="Arg">U</var> or let <var class="Arg">pcp</var> be a pcp of a normal elementary of free abelian subfactor of <var class="Arg">U</var>. Then this function creates a record which can be used as input for the cohomology computations.</p>
<p>The returned cohomology record <var class="Arg">C</var> contains the following entries:</p>
<dl>
<dt><strong class="Mark"><var class="Arg">factor</var></strong></dt>
<dd><p>a pcp of the acting group. If the module is external, then this is <var class="Arg">Pcp(U)</var>. If the module is internal, then this is <var class="Arg">Pcp(U, A)</var> or <var class="Arg">Pcp(U, GroupOfPcp(pcp))</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">mats</var>, <var class="Arg">invs</var> and <var class="Arg">one</var></strong></dt>
<dd><p>the matrix action of <var class="Arg">factor</var> with acting matrices, their inverses and the identity matrix.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">dim</var> and <var class="Arg">char</var></strong></dt>
<dd><p>the dimension and characteristic of the matrices.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">relators</var> and <var class="Arg">enumrels</var></strong></dt>
<dd><p>the right hand sides of the polycyclic relators of <var class="Arg">factor</var> as generator exponents lists and a description for the corresponding left hand sides.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">central</var></strong></dt>
<dd><p>is true, if the matrices <var class="Arg">mats</var> are all trivial. This is used locally for efficiency reasons.</p>
</dd>
</dl>
<p>And additionally, if <span class="SimpleMath">C</span> defines an internal module, then it contains:</p>
<dl>
<dt><strong class="Mark"><var class="Arg">group</var></strong></dt>
<dd><p>the original group <var class="Arg">U</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">normal</var></strong></dt>
<dd><p>this is either <var class="Arg">Pcp(A)</var> or the input <var class="Arg">pcp</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">extension</var></strong></dt>
<dd><p>information on the extension of <var class="Arg">A</var> by <var class="Arg">U/A</var>.</p>
</dd>
</dl>
<p><a id="X874759D582393441" name="X874759D582393441"></a></p>
<h4>8.2 <span class="Heading">Cohomology groups</span></h4>
<p>Let <span class="SimpleMath">U</span> be a pcp-group and <span class="SimpleMath">A</span> a free or elementary abelian pcp-group and a <span class="SimpleMath">U</span>-module. By <span class="SimpleMath">Z^i(U, A)</span> be denote the group of <span class="SimpleMath">i</span>-th cocycles and by <span class="SimpleMath">B^i(U, A)</span> the <span class="SimpleMath">i</span>-th coboundaries. The factor <span class="SimpleMath">Z^i(U,A) / B^i(U,A)</span> is the <span class="SimpleMath">i</span>-th cohomology group. Since <span class="SimpleMath">A</span> is elementary or free abelian, the groups <span class="SimpleMath">Z^i(U, A)</span> and <span class="SimpleMath">B^i(U, A)</span> are elementary or free abelian groups as well.</p>
<p>The <strong class="pkg">Polycyclic</strong> package provides methods to compute first and second cohomology group for a polycyclic group <var class="Arg">U</var>. We write all involved groups additively and we use an explicit description by bases for them. Let <span class="SimpleMath">C</span> be the cohomology record corresponding to <span class="SimpleMath">U</span> and <span class="SimpleMath">A</span>.</p>
<p>Let <span class="SimpleMath">f_1, ..., f_n</span> be the elements in the entry <span class="SimpleMath">factor</span> of the cohomology record <span class="SimpleMath">C</span>. Then we use the following embedding of the first cocycle group to describe 1-cocycles and 1-coboundaries: <span class="SimpleMath">Z^1(U, A) -> A^n : δ ↦ (δ(f_1), ..., δ(f_n))</span></p>
<p>For the second cohomology group we recall that each element of <span class="SimpleMath">Z^2(U, A)</span> defines an extension <span class="SimpleMath">H</span> of <span class="SimpleMath">A</span> by <span class="SimpleMath">U</span>. Thus there is a pc-presentation of <span class="SimpleMath">H</span> extending the pc-presentation of <span class="SimpleMath">U</span> given by the record <span class="SimpleMath">C</span>. The extended presentation is defined by tails in <span class="SimpleMath">A</span>; that is, each relator in the record entry <span class="SimpleMath">relators</span> is extended by an element of <span class="SimpleMath">A</span>. The concatenation of these tails yields a vector in <span class="SimpleMath">A^l</span> where <span class="SimpleMath">l</span> is the length of the record entry <span class="SimpleMath">relators</span> of <span class="SimpleMath">C</span>. We use these tail vectors to describe <span class="SimpleMath">Z^2(U, A)</span> and <span class="SimpleMath">B^2(U, A)</span>. Note that this description is dependent on the chosen presentation in <span class="SimpleMath">C</span>. However, the factor <span class="SimpleMath">Z^2(U, A)/ B^2(U, A)</span> is independent of the chosen presentation.</p>
<p>The following functions are available to compute explicitly the first and second cohomology group as described above.</p>
<p><a id="X85EF170387D39D4A" name="X85EF170387D39D4A"></a></p>
<h5>8.2-1 OneCoboundariesCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCoboundariesCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCocyclesCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwoCoboundariesCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwoCocyclesCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCohomologyCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwoCohomologyCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first 4 functions return bases of the corresponding group. The last 2 functions need to describe a factor of additive abelian groups. They return the following descriptions for these factors.</p>
<dl>
<dt><strong class="Mark"><var class="Arg">gcc</var></strong></dt>
<dd><p>the basis of the cocycles of <var class="Arg">C</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">gcb</var></strong></dt>
<dd><p>the basis of the coboundaries of <var class="Arg">C</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">factor</var></strong></dt>
<dd><p>a description of the factor of cocycles by coboundaries. Usually, it would be most convenient to use additive mappings here. However, these are not available in case that <var class="Arg">A</var> is free abelian and thus we use a description of this additive map as record. This record contains</p>
<dl>
<dt><strong class="Mark"><var class="Arg">gens</var></strong></dt>
<dd><p>a base for the image.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">rels</var></strong></dt>
<dd><p>relative orders for the image.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">imgs</var></strong></dt>
<dd><p>the images for the elements in <var class="Arg">gcc</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">prei</var></strong></dt>
<dd><p>preimages for the elements in <var class="Arg">gens</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">denom</var></strong></dt>
<dd><p>the kernel of the map; that is, another basis for <var class="Arg">gcb</var>.</p>
</dd>
</dl>
</dd>
</dl>
<p>There is an additional function which can be used to compute the second cohomology group over an arbitrary finitely generated abelian group. The finitely generated abelian group should be realized as a factor of a free abelian group modulo a lattice. The function is called as</p>
<p><a id="X79B48D697A8A84C8" name="X79B48D697A8A84C8"></a></p>
<h5>8.2-2 TwoCohomologyModCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwoCohomologyModCR</code>( <var class="Arg">C</var>, <var class="Arg">lat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>where <var class="Arg">C</var> is a cohomology record and <var class="Arg">lat</var> is a basis for a sublattice of a free abelian module. The output format is the same as for <code class="code">TwoCohomologyCR</code>.</p>
<p><a id="X79610E9178BD0C54" name="X79610E9178BD0C54"></a></p>
<h4>8.3 <span class="Heading">Extended 1-cohomology</span></h4>
<p>In some cases more information on the first cohomology group is of interest. In particular, if we have an internal module given and we want to compute the complements using the first cohomology group, then we need additional information. This extended version of first cohomology is obtained by the following functions.</p>
<p><a id="X7E87E3EA81C84621" name="X7E87E3EA81C84621"></a></p>
<h5>8.3-1 OneCoboundariesEX</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCoboundariesEX</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a record consisting of the entries</p>
<dl>
<dt><strong class="Mark"><var class="Arg">basis</var></strong></dt>
<dd><p>a basis for <span class="SimpleMath">B^1(U, A) ≤ A^n</span>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">transf</var></strong></dt>
<dd><p>There is a derivation mapping from <span class="SimpleMath">A</span> to <span class="SimpleMath">B^1(U,A)</span>. This mapping is described here as transformation from <span class="SimpleMath">A</span> to <var class="Arg">basis</var>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">fixpts</var></strong></dt>
<dd><p>the fixpoints of <span class="SimpleMath">A</span>. This is also the kernel of the derivation mapping.</p>
</dd>
</dl>
<p><a id="X8111D2087C16CC0C" name="X8111D2087C16CC0C"></a></p>
<h5>8.3-2 OneCocyclesEX</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCocyclesEX</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a record consisting of the entries</p>
<dl>
<dt><strong class="Mark"><var class="Arg">basis</var></strong></dt>
<dd><p>a basis for <span class="SimpleMath">Z^1(U, A) ≤ A^n</span>.</p>
</dd>
<dt><strong class="Mark"><var class="Arg">transl</var></strong></dt>
<dd><p>a special solution. This is only of interest in case that <span class="SimpleMath">C</span> is an internal module and in this case it gives the translation vector in <span class="SimpleMath">A^n</span> used to obtain complements corresponding to the elements in <span class="SimpleMath">basis</span>. If <span class="SimpleMath">C</span> is not an internal module, then this vector is always the zero vector.</p>
</dd>
</dl>
<p><a id="X84718DDE792FB212" name="X84718DDE792FB212"></a></p>
<h5>8.3-3 OneCohomologyEX</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCohomologyEX</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the combined information on the first cohomology group.</p>
<p><a id="X853E51787A24AE00" name="X853E51787A24AE00"></a></p>
<h4>8.4 <span class="Heading">Extensions and Complements</span></h4>
<p>The natural applications of first and second cohomology group is the determination of extensions and complements. Let <span class="SimpleMath">C</span> be a cohomology record.</p>
<p><a id="X7DA9162085058006" name="X7DA9162085058006"></a></p>
<h5>8.4-1 ComplementCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementCR</code>( <var class="Arg">C</var>, <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the complement corresponding to the 1-cocycle <var class="Arg">c</var>. In the case that <var class="Arg">C</var> is an external module, we construct the split extension of <span class="SimpleMath">U</span> with <span class="SimpleMath">A</span> first and then determine the complement. In the case that <var class="Arg">C</var> is an internal module, the vector <var class="Arg">c</var> must be an element of the affine space corresponding to the complements as described by <code class="code">OneCocyclesEX</code>.</p>
<p><a id="X7F8984D386A813D6" name="X7F8984D386A813D6"></a></p>
<h5>8.4-2 ComplementsCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementsCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns all complements using the correspondence to <span class="SimpleMath">Z^1(U,A)</span>. Further, this function returns fail, if <span class="SimpleMath">Z^1(U,A)</span> is infinite.</p>
<p><a id="X7FAB3EB0803197FA" name="X7FAB3EB0803197FA"></a></p>
<h5>8.4-3 ComplementClassesCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementClassesCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns complement classes using the correspondence to <span class="SimpleMath">H^1(U,A)</span>. Further, this function returns fail, if <span class="SimpleMath">H^1(U,A)</span> is infinite.</p>
<p><a id="X8759DC59799DD508" name="X8759DC59799DD508"></a></p>
<h5>8.4-4 ComplementClassesEfaPcps</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementClassesEfaPcps</code>( <var class="Arg">U</var>, <var class="Arg">N</var>, <var class="Arg">pcps</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">N</span> be a normal subgroup of <span class="SimpleMath">U</span>. This function returns the complement classes to <span class="SimpleMath">N</span> in <span class="SimpleMath">U</span>. The classes are computed by iteration over the <span class="SimpleMath">U</span>-invariant efa series of <span class="SimpleMath">N</span> described by <var class="Arg">pcps</var>. If at some stage in this iteration infinitely many complements are discovered, then the function returns fail. (Even though there might be only finitely many conjugacy classes of complements to <span class="SimpleMath">N</span> in <span class="SimpleMath">U</span>.)</p>
<p><a id="X7B0EC76D81A056AB" name="X7B0EC76D81A056AB"></a></p>
<h5>8.4-5 ComplementClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementClasses</code>( [<var class="Arg">V</var>, ]<var class="Arg">U</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">N</span> and <span class="SimpleMath">U</span> be normal subgroups of <span class="SimpleMath">V</span> with <span class="SimpleMath">N ≤ U ≤ V</span>. This function attempts to compute the <span class="SimpleMath">V</span>-conjugacy classes of complements to <span class="SimpleMath">N</span> in <span class="SimpleMath">U</span>. The algorithm proceeds by iteration over a <span class="SimpleMath">V</span>-invariant efa series of <span class="SimpleMath">N</span>. If at some stage in this iteration infinitely many complements are discovered, then the algorithm returns fail.</p>
<p><a id="X85F3B55C78CF840B" name="X85F3B55C78CF840B"></a></p>
<h5>8.4-6 ExtensionCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtensionCR</code>( <var class="Arg">C</var>, <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the extension corresponding to the 2-cocycle <span class="SimpleMath">c</span>.</p>
<p><a id="X81DC85907E0948FD" name="X81DC85907E0948FD"></a></p>
<h5>8.4-7 ExtensionsCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtensionsCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns all extensions using the correspondence to <span class="SimpleMath">Z^2(U,A)</span>. Further, this function returns fail, if <span class="SimpleMath">Z^2(U,A)</span> is infinite.</p>
<p><a id="X7AE16E3687E14B24" name="X7AE16E3687E14B24"></a></p>
<h5>8.4-8 ExtensionClassesCR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtensionClassesCR</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns extension classes using the correspondence to <span class="SimpleMath">H^2(U,A)</span>. Further, this function returns fail, if <span class="SimpleMath">H^2(U,A)</span> is infinite.</p>
<p><a id="X7986997B78AD3292" name="X7986997B78AD3292"></a></p>
<h5>8.4-9 SplitExtensionPcpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SplitExtensionPcpGroup</code>( <var class="Arg">U</var>, <var class="Arg">mats</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the split extension of <var class="Arg">U</var> by the <span class="SimpleMath">U</span>-module described by <var class="Arg">mats</var>.</p>
<p><a id="X823771527DBD857D" name="X823771527DBD857D"></a></p>
<h4>8.5 <span class="Heading">Constructing pcp groups as extensions</span></h4>
<p>This section contains an example application of the second cohomology group to the construction of pcp groups as extensions. The following constructs extensions of the group of upper unitriangular matrices with its natural lattice.</p>
<div class="example"><pre>
# get the group and its matrix action
<span class="GAPprompt">gap></span> <span class="GAPinput">G := UnitriangularPcpGroup(3,0);</span>
Pcp-group with orders [ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mats := G!.mats;</span>
[ [ [ 1, 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],
[ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]
# set up the cohomology record
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CRRecordByMats(G,mats);;</span>
# compute the second cohomology group
<span class="GAPprompt">gap></span> <span class="GAPinput">cc := TwoCohomologyCR(C);;</span>
# the abelian invariants of H^2(G,M)
<span class="GAPprompt">gap></span> <span class="GAPinput">cc.factor.rels;</span>
[ 2, 0, 0 ]
# construct an extension which corresponds to a cocycle that has
# infinite image in H^2(G,M)
<span class="GAPprompt">gap></span> <span class="GAPinput">c := cc.factor.prei[2];</span>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H := ExtensionCR( CR, c);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]
# check that the extension does not split - get the normal subgroup
<span class="GAPprompt">gap></span> <span class="GAPinput">N := H!.module;</span>
Pcp-group with orders [ 0, 0, 0 ]
# create the interal module
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CRRecordBySubgroup(H,N);;</span>
# use the complements routine
<span class="GAPprompt">gap></span> <span class="GAPinput">ComplementClassesCR(C);</span>
[ ]
</pre></div>
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