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<h1>GAP 4 Package <strong class="pkg">FinInG</strong></h1>


<h2>Finite Incidence Geometry</h2>

<p>1.5.6</p>

<p>27/07/2023</p>

</div>
<p><b>John Bamberg
    </b>
</p><p><b>Anton Betten
    </b>
</p><p><b>Jan De Beule
    </b>
</p><p><b>Philippe Cara
    </b>
</p><p><b>Michel Lavrauw
    </b>
</p><p><b>Max Neunhöffer
    </b>
</p><p><b>
    
    
    </b>
<br />Email: <span class="URL"><a href="mailto:support@fining.org">support@fining.org</a></span>
<br />Homepage: <span class="URL"><a href="http://www.fining.org">http://www.fining.org</a></span>
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2014-2022 by the authors</p>

<p>This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.</p>

<p>The development group of <strong class="pkg">FinInG</strong> welcomes contact with users. In case you have obtained the package as a deposited package part of archive during the installation of GAP, we call on your beneficence to register at <span class="URL"><a href="http://www.fining.org">http://www.fining.org</a></span> when you use <strong class="pkg">FinInG</strong>, or to tell us by sending an e-mail to <span class="URL"><a href="mailto:council@fining.org">council@fining.org</a></span>.</p>

<p>Please also tell us about the use of <strong class="pkg">FinInG</strong> in your research or teaching. We are very interested in results obtained using <strong class="pkg">FinInG</strong> and we might refer to your work in the future. If your work is published, we ask you to cite <strong class="pkg">FinInG</strong> like a journal article or book. We provide the necessary BibTex and LaTeX code in Section <a href="chap1.html#X837E428E80B6049C"><span class="RefLink">1.2</span></a>.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>The development phase of <strong class="pkg">FinInG</strong> started roughly in 2005. The idea to write a package for projective geometry in GAP had emerged before, and resulted in <strong class="pkg">pg</strong>, a relic that still can be found in the undeposited packages section of the GAP website. One of the main objectives was to develop the new package was to create a tighter connection between finite geometry and the group theoretical functions present in GAP.</p>

<p>The authors thank Michael Pauley, Maska Law and Sven Reichard, for their contributions during the early days of the project.</p>

<p>Jan De Beule and Michel Lavrauw have been supported by the Research Foundation Flanders -- Belgium (FWO), and John Bamberg has been supported by a Marie Curie grant and an ARC grant during almost the whole development phase of <strong class="pkg">FinInG</strong>. The authors are grateful for this support.</p>

<p>John Bamberg, Philippe Cara and Jan De Beule have spent several weeks in Vicenza while developing <strong class="pkg">FinInG</strong>. We acknowledge the hospitality of Michel Lavrauw and Corrado Zanella. Our stays in Vicenza were always fruitful and very enjoyable. During or daily morning and afternoon coffee breaks, we discussed several topics, while enjoying coffee in <em>Caffe Pigaffeta</em>. Although one cannot acknowledge everybody, such as the bakery who provided us with the sandwiches, we acknowledge very much the hospitality of Carla and Luigi, who introduced us to many different kinds of coffees from all over the world.</p>

<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X873C99678745ABAF">1.1 <span class="Heading">Philosophy</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X837E428E80B6049C">1.2 <span class="Heading">How to cite <strong class="pkg">FinInG</strong></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X87B5A1377BCABBD6">1.3 <span class="Heading">Overview of this manual</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7EEBA9577BA68BA6">1.4 <span class="Heading">Getting and installing <strong class="pkg">FinInG</strong></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F499E8A79509971">1.4-1 <span class="Heading">Installation procedure under UNIX like systems</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X83A6C48E806EC0E9">1.4-2 <span class="Heading">Compiling packages</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X78F15DAE873084A5">1.4-3 <span class="Heading">Updating <strong class="pkg">FinInG</strong></span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X83BC24CC831A2542">1.5 <span class="Heading">The Development Team</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7A489A5D79DA9E5C">2 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X81660CB279889CB6">2.1 <span class="Heading">Elementary examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8016E6857D53F2ED">2.1-1 <span class="Heading">subspaces of projective spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B99511887D41A95">2.1-2 <span class="Heading">Subspaces of classical polar spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8555398C83677C27">2.1-3 <span class="Heading">Underlying objects</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8771ACB879E479C6">2.1-4 <span class="Heading">Constructing polar spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85D3BB2A8274DDCB">2.1-5 <span class="Heading">Some collineation groups</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X825F78F57E309197">2.2 <span class="Heading">Some objects with interesting combinatorial properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X815BB30986E84DB1">2.2-1 <span class="Heading">The Tits ovoid</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E79F18B8170B4B3">2.2-2 <span class="Heading">Lines meeting a hermitian curve</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85C255FD78C50992">2.2-3 <span class="Heading">The Patterson ovoid</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X80B93785876EF3E0">2.2-4 <span class="Heading">A hyperoval</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X876240A479A5717C">2.3 <span class="Heading">Geometry morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79CE092B7E17DF24">2.3-1 <span class="Heading">Isomorphic polar spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83ADB5AE8624C74C">2.3-2 <span class="Heading">Intertwiners</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C7438AB86A493FE">2.3-3 <span class="Heading">Klein correspondence</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X869EB94D841AE028">2.3-4 <span class="Heading">Embedding in a subspace</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FE8E4BF7E700E65">2.3-5 <span class="Heading">Subgeometries</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X838BBDD97FA03FD0">2.3-6 <span class="Heading">Embedding by field reduction</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X855C8E6D819EB975">2.4 <span class="Heading">Some geometrical objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8475841778D3BEEC">2.4-1 <span class="Heading">Spreads of <span class="SimpleMath">W(5,3)</span></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81F516D07E8165B9">2.4-2 <span class="Heading">Distance-6 spread of the split Cayley hexagon</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F13364A7EEA2AD1">2.5 <span class="Heading">Some particular incidence geometries</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79623B9E7D5816B3">2.5-1 <span class="Heading">The split Cayley hexagon</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8528558E87DE72C5">2.5-2 <span class="Heading">An (apartment of) a building of type <span class="SimpleMath">E_6</span></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B783473852C7899">2.5-3 <span class="Heading">A rank 4 geometry for <span class="SimpleMath">PSL(2,11)</span></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X80128FF17BB62C83">2.5-4 <span class="Heading">The Ree-Tits octagon of order <span class="SimpleMath">[2,4]</span> as coset geometry</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BA462527B2777BC">2.6 <span class="Heading">Elation generalised quadrangles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E3707857A74AB5E">2.6-1 <span class="Heading">The classical q-clan</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83357ED78789111E">2.6-2 <span class="Heading">Two ways to construct a flock generalised quadrangle from a Kantor-Knuth semifield q-clan</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X87EC44BF7F24486E">2.7 <span class="Heading">Algebraic varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7ABCF9637B60FF37">2.7-1 <span class="Heading">A projective variety</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X838ACF8A7F100A2B">3 <span class="Heading">Incidence Geometry</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7FB175337C4F8B76">3.1 <span class="Heading">Incidence structures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81F7D8FA82A55561">3.1-1 IsIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78C74BE87E050E84">3.1-2 IsIncidenceGeometry</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C3258887C8DF5C1">3.1-3 IncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B0347E2863C1E8C">3.1-4 <span class="Heading">Main categories in <code class="code">IsIncidenceGeometry</code></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87FFB1648575FFF2">3.1-5 <span class="Heading">Examples of categories of incidence geometries</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E574AB27DA97063">3.1-6 TypesOfElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X827146F37E2AA841">3.1-7 Rank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X815BE6D57D623452">3.1-8 IncidenceGraph</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7BBDB0AE7E29F3FB">3.2 <span class="Heading">Elements of incidence structures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X827CD3C881DC8364">3.2-1 <span class="Heading">Main categories for individual elements of incidence structures</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X810D4D6D87069697">3.2-2 UnderlyingObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X823B3D0F87FB5403">3.2-3 Type</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7809B7C183FA7213">3.2-4 ObjectToElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DE974E687A2ABFB">3.2-5 <span class="Heading">Main categories for collections of all the elements of a given type of an incidence structure</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87657AEF7E2C50F9">3.2-6 ElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87657AEF7E2C50F9">3.2-7 ElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87E64DA67C3D6661">3.2-8 <span class="Heading">Short names for ElementsOfIncidenceStructure</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86CF041F7FA486D6">3.2-9 NrElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79730D657AB219DB">3.2-10 Random</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A9ED8327C40B445">3.2-11 IsIncident</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X799DB77886B8ABDB">3.2-12 AmbientGeometry</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7DACFB6785029BF0">3.3 <span class="Heading">Flags of incidence structures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E204A78815C46DD">3.3-1 FlagOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A453E0E861F2C94">3.3-2 IsChamberOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7AEFC2C57F10C3A7">3.3-3 IsEmptyFlag</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86FFFBC584B97371">3.3-4 ElementsOfFlag</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X827146F37E2AA841">3.3-5 Rank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X858ADA3B7A684421">3.3-6 Size</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X799DB77886B8ABDB">3.3-7 AmbientGeometry</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X823B3D0F87FB5403">3.3-8 Type</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A9ED8327C40B445">3.3-9 IsIncident</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7AA14EDF7B0B1569">3.4 <span class="Heading">Shadow of elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FFA08DA85C5251C">3.4-1 ShadowOfElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81A8365A7FE68447">3.4-2 ElementsIncidentWithElementOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E86E6417871730C">3.4-3 ShadowOfFlag</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78BE3D727B060301">3.4-4 ResidueOfFlag</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E29C31D7CB5DB23">3.4-5 <span class="Heading">Short names for ElementsIncidentWithElementOfIncidenceStructure</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8133F88478BAFCB7">3.5 <span class="Heading">Enumerating elements of an incidence structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83ADF8287ED0668E">3.5-1 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7EF8910F82B45EC7">3.5-2 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7EBA57FC7CCF8449">3.5-3 List</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8289FCCC8274C89D">3.5-4 AsList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X84D77D437B5F3716">3.6 <span class="Heading">Lie geometries</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D012B9F86E63702">3.6-1 <span class="Heading">Main categories in <code class="code">IsLieGeometry</code></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8606750A8586DF8D">3.6-2 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D544D7985A4572D">3.6-3 UnderlyingVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84FDF25D797B874B">3.6-4 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85DFF5177CA51AE0">3.6-5 IsEmptySubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7FBCF60385E8C1D8">3.7 <span class="Heading">Elements of Lie geometries</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82E9593B8074AECB">3.7-1 VectorSpaceToElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X810D4D6D87069697">3.7-2 UnderlyingObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87BDB89B7AAFE8AD"><code>3.7-3 \in</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X814C3AC27E49AD5B">3.7-4 <span class="Heading">More short names for <code class="file">ElementsIncidentWithElementOfIncidenceStructure</code></span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7A9EBF9782671634">3.8 <span class="Heading">Changing the ambient geometry of elements of a Lie geometry</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8561C0117FD76C94">3.8-1 ElementToElement</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X83BBAA668672A76D">4 <span class="Heading">Projective Spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7862BC887D20B37A">4.1 <span class="Heading">Projective Spaces and basic operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79B440FF7EFBA661">4.1-1 IsProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7962DA507C64FCBA">4.1-2 ProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84FDF25D797B874B">4.1-3 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7BCBA564829D9E89">4.1-4 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D544D7985A4572D">4.1-5 UnderlyingVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8606750A8586DF8D">4.1-6 AmbientSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8016E6857D53F2ED">4.2 <span class="Heading">Subspaces of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82E9593B8074AECB">4.2-1 VectorSpaceToElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8461BCEF862B9A7B">4.2-2 EmptySubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84FDF25D797B874B">4.2-3 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87657AEF7E2C50F9">4.2-4 ElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87E64DA67C3D6661">4.2-5 <span class="Heading">Short names for ElementsOfIncidenceStructure</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7904128479BDFCC9">4.2-6 <span class="Heading">Incidence and containment</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X870D9D9A7F11806F">4.2-7 StandardFrame</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84E3985A8700B302">4.2-8 Coordinates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CE4FD76820B503A">4.2-9 DualCoordinatesOfHyperplane</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86628227863989E5">4.2-10 HyperplaneByDualCoordinates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X801A95907B13F447">4.2-11 EquationOfHyperplane</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8606750A8586DF8D">4.2-12 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7BCBA564829D9E89">4.2-13 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79730D657AB219DB">4.2-14 Random</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D3C5D3B7AA4DE28">4.2-15 RandomSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X875BE2957FAF6209">4.2-16 Span</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8469B54180FE1E4C">4.2-17 Meet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E204A78815C46DD">4.2-18 FlagOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AEFC2C57F10C3A7">4.2-19 IsEmptyFlag</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A453E0E861F2C94">4.2-20 IsChamberOfIncidenceStructure</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BD8312C85784503">4.3 <span class="Heading">Shadows of Projective Subspaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FFA08DA85C5251C">4.3-1 ShadowOfElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E86E6417871730C">4.3-2 ShadowOfFlag</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81A8365A7FE68447">4.3-3 ElementsIncidentWithElementOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E29C31D7CB5DB23">4.3-4 <span class="Heading">Short names for <code class="file">ElementsIncidentWithElementOfIncidenceStructure</code></span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X799F3A2A86F82E5B">4.4 <span class="Heading">Enumerating subspaces of a projective space</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83ADF8287ED0668E">4.4-1 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EF8910F82B45EC7">4.4-2 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EBA57FC7CCF8449">4.4-3 List</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X816FCFB683915E8A">5 <span class="Heading">Projective Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7A9762F8861B0772">5.1 <span class="Heading"> Projectivities, collineations and correlations of projective spaces.</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X851186297A91C1C6">5.1-1 <span class="Heading">Categories for group elements</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BBF688083857760">5.1-2 <span class="Heading">Representations for group elements</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8160615081358132">5.1-3 <span class="Heading">Projectivities</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E881C237D117C6C">5.1-4 <span class="Heading">Collineations of projective spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B89B51F86AE2BCC">5.1-5 <span class="Heading">Projective strictly semilinear maps</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X815B68277D0500C3">5.1-6 <span class="Heading">Correlations and collineations</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X78EDF0357B58FC0E">5.2 <span class="Heading">Construction of projectivities, collineations and correlations.</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X877DA4E185A1D9C7">5.2-1 Projectivity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7AB452B2781EF128">5.2-2 CollineationOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X81ED446485A71588">5.2-3 ProjectiveSemilinearMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80649C427E3BCBFF">5.2-4 IdentityMappingOfElementsOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X841607A77B841CC9">5.2-5 StandardDualityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X78BBC4E27B2E06D6">5.2-6 CorrelationOfProjectiveSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X83A5F86F82598AA6">5.3 <span class="Heading">Basic operations for projectivities, collineations and correlations of
projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X865507568182424E">5.3-1 Representative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CA72CB07E3122F1">5.3-2 MatrixOfCollineation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E8B65547970F689">5.3-3 MatrixOfCorrelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BCBA564829D9E89">5.3-4 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B60C4257C46ED4B">5.3-5 FieldAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7FE527AB81C2B675">5.3-6 ProjectiveSpaceIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X84F59A2687C62763">5.3-7 Order</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X78E99D9086D64FD9">5.4 <span class="Heading">The groups PΓL, PGL, and PSL in <strong class="pkg">FinInG</strong>
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X850A954887CA9A55">5.4-1 ProjectivityGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83FF6FA0790D5747">5.4-2 CollineationGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BE9CE127ACFA6C2">5.4-3 SpecialProjectivityGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DEA3BDA82C7B855">5.4-4 IsProjectivityGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B1FC1327FD85D18">5.4-5 IsCollineationGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X81444EF57E228232">5.4-6 CorrelationCollineationGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7C4C7ADE8746C1B1">5.5 <span class="Heading">Basic operations for projective groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BCBA564829D9E89">5.5-1 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E6926C6850E7C4E">5.5-2 Dimension</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X87327CBC857D6801">5.6 <span class="Heading">Natural embedding of a collineation group in a correlation/collineation group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86452F8587CBAEA0">5.6-1 Embedding</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7AAD7DDD7E19595E">5.7 <span class="Heading">Basic action of projective group elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D21FB1A7D21FB1A"><code>5.7-1 \^</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7EBA895D7A501CE0">5.8 <span class="Heading">Projective group actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X84A3D5357872EC3B">5.8-1 OnProjSubspaces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X798053D47D8187EC">5.8-2 ActionOnAllProjPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86B4C03E85ADD0C2">5.8-3 OnProjSubspacesExtended</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X809F0F2B857FA178">5.9 <span class="Heading">Special subgroups of the projectivity group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86FF1DDE8356E966">5.9-1 ElationOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E5660A17A4B1349">5.9-2 ProjectiveElationGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86319DCD7AF98E28">5.9-3 HomologyOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X82FBABF384960A3D">5.9-4 ProjectiveHomologyGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7FFD731684606BC6">5.10 <span class="Heading">Nice Monomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7965086E82ABCF41">5.10-1 NiceMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B47BE0983E93A83">5.10-2 NiceObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CE11961817B311C">5.10-3 FINING</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B102DAE7E0CCF47">5.10-4 CanComputeActionOnPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D8C4B657FD6F7BA">5.10-5 NiceMonomorphismByDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BCBD96786901FF9">5.10-6 NiceMonomorphismByOrbit</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6.html#X87BA55CB86B110EC">6 <span class="Heading">Polarities of Projective Spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X86D948C3875A5005">6.1 <span class="Heading">Creating polarities of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7EBD8C07802562B7">6.1-1 PolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7EBD8C07802562B7">6.1-2 PolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7EBD8C07802562B7">6.1-3 PolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7EBD8C07802562B7">6.1-4 PolarityOfProjectiveSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X81CC3CBE7879FD7B">6.2 <span class="Heading">Operations, attributes and properties for polarities of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X793BE1A27BF349F3">6.2-1 SesquilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7BCBA564829D9E89">6.2-2 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X847AFB4C81A90B3F">6.2-3 GramMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7C55F56E7E34768B">6.2-4 CompanionAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7C15F40A85F167F4">6.2-5 IsHermitianPolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X855077387D144CDE">6.2-6 IsSymplecticPolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87FFEEAE7FC2EE41">6.2-7 IsOrthogonalPolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8372FCBC8313572F">6.2-8 IsPseudoPolarityOfProjectiveSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X83F8149B7D23301E">6.3 <span class="Heading">Polarities, absolute points, totally isotropic elements and finite classical polar
spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X81C291357E5B2408">6.3-1 GeometryOfAbsolutePoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8686AB4D798970BC">6.3-2 AbsolutePoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X863BC8E57C98A471">6.3-3 PolarSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7ADFEAC07CE25530">6.4 <span class="Heading">Commuting polarities</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap7.html#X7F96B1327C022A28">7 <span class="Heading">Finite Classical Polar Spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7F96B1327C022A28">7.1 <span class="Heading">Finite Classical Polar Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83A92A177BE4E0BB">7.1-1 IsClassicalPolarSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X863BC8E57C98A471">7.1-2 PolarSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X850CD32686B0656B">7.2 <span class="Heading">Canonical and standard Polar Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X850980EF8607BDA7">7.2-1 SymplecticSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A8CF840833D3D8B">7.2-2 HermitianPolarSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7CEDD5CC7A6B0E64">7.2-3 ParabolicQuadric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7CF248A2864BCD8F">7.2-4 HyperbolicQuadric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85EEF70480D7149E">7.2-5 EllipticQuadric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F6D25D57BC697B5">7.2-6 IsCanonicalPolarSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8240380A7F8B6EC3">7.2-7 CanonicalPolarSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7EE25EDF7ADC30D7">7.2-8 StandardPolarSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7A04340A7EC9215B">7.3 <span class="Heading">Basic operations for finite classical polar spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D544D7985A4572D">7.3-1 UnderlyingVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8606750A8586DF8D">7.3-2 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X84FDF25D797B874B">7.3-3 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X827146F37E2AA841">7.3-4 Rank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7BCBA564829D9E89">7.3-5 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8412E6307ACC82C2">7.3-6 IsHyperbolicQuadric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85386DAB8056E316">7.3-7 IsEllipticQuadric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87ADB8B5790776C7">7.3-8 IsParabolicQuadric</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X787E0AEA8284B34B">7.4 <span class="Heading">Subspaces of finite classical polar spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X82E9593B8074AECB">7.4-1 VectorSpaceToElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8461BCEF862B9A7B">7.4-2 EmptySubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X84FDF25D797B874B">7.4-3 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87657AEF7E2C50F9">7.4-4 ElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8606750A8586DF8D">7.4-5 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X84E3985A8700B302">7.4-6 Coordinates</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X8472E78A79F44828">7.5 <span class="Heading">Basic operations for polar spaces and subspaces of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7904128479BDFCC9">7.5-1 <span class="Heading">Incidence and containment</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X875BE2957FAF6209">7.5-2 Span</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8469B54180FE1E4C">7.5-3 Meet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8176044E7D2242D9">7.5-4 IsCollinear</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7EBD8C07802562B7">7.5-5 PolarityOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X863F1924805F6C63">7.5-6 TypeOfSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X82C38639870C0DD6">7.5-7 TangentSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8520DC538717C6BE">7.5-8 Pole</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X808AB7B9840ABC27">7.5-9 EvaluateForm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7AA14EDF7B0B1569">7.6 <span class="Heading">Shadow of elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7FFA08DA85C5251C">7.6-1 ShadowOfElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81A8365A7FE68447">7.6-2 ElementsIncidentWithElementOfIncidenceStructure</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7988AF9978E75E37">7.7 <span class="Heading">Projective Orthogonal/Unitary/Symplectic groups in <strong class="pkg">FinInG</strong></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X857C261585038CB8">7.7-1 SpecialIsometryGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87708229834E4C44">7.7-2 IsometryGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X812836E48364AE6D">7.7-3 SimilarityGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83FF6FA0790D5747">7.7-4 CollineationGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X855D48A07E0BBCDB">7.8 <span class="Heading">Enumerating subspaces of polar spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7AB1BA95825BDE71">7.8-1 <span class="Heading">Enumerators for polar spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7EF8910F82B45EC7">7.8-2 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X861463147B738DF1">7.8-3 <span class="Heading">Iterators for polar spaces</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83ADF8287ED0668E">7.8-4 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8289FCCC8274C89D">7.8-5 AsList</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap8.html#X87A0A15D8588D62F">8 <span class="Heading">Orbits, stabilisers and actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X81E0FF0587C54543">8.1 <span class="Heading">Orbits</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7CF168A0818C553D">8.1-1 FiningOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X827CD5727A7DA4A3">8.1-2 FiningOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X822D1CFF7DDC11D7">8.1-3 FiningOrbitsDomain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7EAB52F67B3A0003">8.2 <span class="Heading">Stabilisers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X83B7F30F81A0BCE3">8.2-1 FiningStabiliser</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X854826AC7DA9FB6B">8.2-2 FiningStabiliserOrb</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7929900F79B6000C">8.2-3 FiningSetwiseStabiliser</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7A625B4D7992EEBC">8.2-4 StabiliserGroupOfSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7F50965D7B9C1BC5">8.2-5 ProjectiveStabiliserGroupOfSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X81B7F825830CD669">8.2-6 SpecialProjectiveStabiliserGroupOfSubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7B449F3B7F23A30A">8.3 <span class="Heading">Actions and nice monomorphisms revisited</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X86A646FF8668D82E">8.3-1 <span class="Heading">Action functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X8474367181BB501E">8.3-2 <span class="Heading">Generic GAP functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7965086E82ABCF41">8.3-3 NiceMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7B47BE0983E93A83">8.3-4 NiceObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X86AC831981D89DF1">8.3-5 <span class="Heading">Different behaviour for different collineation groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X80F4EDD487059D70">8.3-6 SetParent</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap9.html#X7A63E8817A819046">9 <span class="Heading">Affine Spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7ADF809E85917970">9.1 <span class="Heading">Affine spaces and basic operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X793755BF8775B629">9.1-1 IsAffineSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87C360337E16E6CC">9.1-2 AffineSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E6926C6850E7C4E">9.1-3 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7BCBA564829D9E89">9.1-4 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7D544D7985A4572D">9.1-5 UnderlyingVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8606750A8586DF8D">9.1-6 AmbientSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7AC346337E23D34F">9.2 <span class="Heading">Subspaces of affine spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X82DE73E884CF8172">9.2-1 AffineSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87657AEF7E2C50F9">9.2-2 ElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87E64DA67C3D6661">9.2-3 <span class="Heading">Short names for ElementsOfIncidenceStructure</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7904128479BDFCC9">9.2-4 <span class="Heading">Incidence and containment</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8606750A8586DF8D">9.2-5 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7BCBA564829D9E89">9.2-6 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X875BE2957FAF6209">9.2-7 Span</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8469B54180FE1E4C">9.2-8 Meet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7FE5611C7A4C0FA2">9.2-9 IsParallel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7A9F3647866EEEE3">9.2-10 ParallelClass</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X835B9A1F7EFE4640">9.3 <span class="Heading">Shadows of Affine Subspaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7FFA08DA85C5251C">9.3-1 ShadowOfElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E86E6417871730C">9.3-2 ShadowOfFlag</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7836304580E12428">9.4 <span class="Heading">Iterators and enumerators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83ADF8287ED0668E">9.4-1 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7EF8910F82B45EC7">9.4-2 Enumerator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X78B78D517B22FB7E">9.5 <span class="Heading">Affine groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8388486C80D0991D">9.5-1 AffineGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83FF6FA0790D5747">9.5-2 CollineationGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E819F357EEEF31C">9.5-3 OnAffineSpaces</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X8769AA7080854675">9.6 <span class="Heading">Low level operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X85C1B3067C6014D2">9.6-1 IsVectorSpaceTransversal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8210A9FA7F9FA6DF">9.6-2 VectorSpaceTransversal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7B2163767BCA3CC6">9.6-3 VectorSpaceTransversalElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8292D00E82069C27">9.6-4 ComplementSpace</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap10.html#X876240A479A5717C">10 <span class="Heading">Geometry Morphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X850559BF7886E0D2">10.1 <span class="Heading">Geometry morphisms in FinInG</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7A3BDDBE7BADCF0A">10.1-1 IsGeometryMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X86E0B2117DD536D2">10.1-2 Intertwiner</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7926E5367D0C80B7">10.2 <span class="Heading">Type preserving bijective geometry morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8774AF5B840E488D">10.2-1 IsomorphismPolarSpaces</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X79C677CD7B7EC451">10.3 <span class="Heading">Klein correspondence and derived dualities</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7C8E719883F407BB">10.3-1 PluckerCoordinates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7CD8E5187C5B7116">10.3-2 KleinCorrespondence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7CD8E5187C5B7116">10.3-3 KleinCorrespondence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7FE537C57F2F6BAA">10.3-4 KleinCorrespondenceExtended</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8472EBA584544B51">10.3-5 NaturalDuality</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7954DA578247B06F">10.3-6 SelfDuality</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X86D21DCB7C0029F9">10.4 <span class="Heading">Embeddings of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X857976B679B8EC30">10.4-1 NaturalEmbeddingBySubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X82510EF078F0806E">10.4-2 NaturalEmbeddingBySubField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7BC7FCDC7D9E1A09">10.4-3 <span class="Heading"> Embedding of projective spaces by field reduction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7F84986879DD952A">10.4-4 BlownUpSubspaceOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D2D9AD287532FDB">10.4-5 NaturalEmbeddingByFieldReduction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7C00DD48787B1EEE">10.5 <span class="Heading">Embeddings of polar spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X857976B679B8EC30">10.5-1 NaturalEmbeddingBySubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X82510EF078F0806E">10.5-2 NaturalEmbeddingBySubField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7823BA95797898CE">10.5-3 <span class="Heading">Embedding of polar spaces by field reduction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D2D9AD287532FDB">10.5-4 NaturalEmbeddingByFieldReduction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D2D9AD287532FDB">10.5-5 NaturalEmbeddingByFieldReduction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X81FAC1DE7C4B1972">10.6 <span class="Heading">Projections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7B9E8AB682EC07C2">10.6-1 NaturalProjectionBySubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7952EE1A80D53825">10.7 <span class="Heading">Projective completion</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7E23CA137AEFD1D2">10.7-1 ProjectiveCompletion</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap11.html#X87EC44BF7F24486E">11 <span class="Heading">Algebraic Varieties</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X87EC44BF7F24486E">11.1 <span class="Heading">Algebraic Varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X85B05D0385B6032A">11.1-1 AlgebraicVariety</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7D9C2FA47BCDEBDB">11.1-2 DefiningListOfPolynomials</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X8606750A8586DF8D">11.1-3 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X86605DF87C562635">11.1-4 PointsOfAlgebraicVariety</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X83ADF8287ED0668E">11.1-5 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X87BDB89B7AAFE8AD"><code>11.1-6 \in</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X79EC6F8381337C08">11.2 <span class="Heading">Projective Varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X83D5414481C9E1D0">11.2-1 ProjectiveVariety</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X8030D25C79C50847">11.3 <span class="Heading">Quadrics and Hermitian varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7E233D1979CEB18B">11.3-1 HermitianVariety</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X850EDF127B1A2C4D">11.3-2 QuadraticVariety</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X8421249486CEE1A0">11.3-3 QuadraticForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X793BE1A27BF349F3">11.3-4 SesquilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X863BC8E57C98A471">11.3-5 PolarSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X82BE5DEE843F5490">11.4 <span class="Heading">Affine Varieties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7A0D03A37AEF48B3">11.4-1 AffineVariety</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X862822D57D48DD8E">11.5 <span class="Heading">Geometry maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7DE8173F80E07AB1">11.5-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X829F76BB80BD55DB">11.5-2 Range</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7CFAB0157BFB1806">11.5-3 ImageElm</a></span>
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