Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/Sources/formale Sprachen/GAP/pkg/fining/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 27.6.2023 mit Größe 74 kB

Quelle chap13.html

Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/fining/doc/chap13.html

<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (FinInG) - Chapter 13: Coset Geometries and Diagrams</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap13" onload="jscontent()">

<div class="chlinktop">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapC.html">C</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap12.html">[Previous Chapter]</a> <a href="chap14.html">[Next Chapter]</a> </div>

<a href="chap13_mj.html">[MathJax on]</a>
<a id="X8328AFAC7CF1EB1B" name="X8328AFAC7CF1EB1B"></a>
<div class="ChapSects"><a href="chap13.html#X8328AFAC7CF1EB1B">13 Coset Geometries and Diagrams</a>
<div class="ContSect"> <a href="chap13.html#X781B20AC8097AC9F">13.1 Coset Geometries</a>

<div class="ContSSBlock">
 <a href="chap13.html#X877735BF810DEAD5">13.1-1 IsCosetGeometry</a>
 <a href="chap13.html#X86C97C4780C27E6F">13.1-2 CosetGeometry</a>
 <a href="chap13.html#X7A9ED8327C40B445">13.1-3 IsIncident</a>
 <a href="chap13.html#X86CC6815824E71E4">13.1-4 ParabolicSubgroups</a>
 <a href="chap13.html#X824D5D558425B4B5">13.1-5 AmbientGroup</a>
 <a href="chap13.html#X7E5B687E86068196">13.1-6 Borelsubgroup</a>
 <a href="chap13.html#X82FB0B387E53B073">13.1-7 RandomElement</a>
 <a href="chap13.html#X87E9C0D3799F95BA">13.1-8 RandomFlag</a>
 <a href="chap13.html#X82DA90817A70103A">13.1-9 RandomChamber</a>
 <a href="chap13.html#X7FEFCB907CAAA9B1">13.1-10 IsFlagTransitiveGeometry</a>
 <a href="chap13.html#X7C0E5EDD806CE4C1">13.1-11 OnCosetGeometryElement</a>
 <a href="chap13.html#X7D21FB1A7D21FB1A"><code>13.1-12 \^</code></a>
 <a href="chap13.html#X7D21FB1A7D21FB1A"><code>13.1-13 \^</code></a>
 <a href="chap13.html#X7F74FD648751B7E6">13.1-14 IsFirmGeometry</a>
 <a href="chap13.html#X7F8A17357F9FFA00">13.1-15 IsThickGeometry</a>
 <a href="chap13.html#X830213C3843A1BA2">13.1-16 IsThinGeometry</a>
 <a href="chap13.html#X877F178084D9A589">13.1-17 IsConnected</a>
 <a href="chap13.html#X83A5107981824C2F">13.1-18 IsResiduallyConnected</a>
 <a href="chap13.html#X80E7BC8F8612176D">13.1-19 StandardFlagOfCosetGeometry</a>
 <a href="chap13.html#X875E89FF7ECD44A6">13.1-20 FlagToStandardFlag</a>
 <a href="chap13.html#X7F3A2B47833EA22F">13.1-21 CanonicalResidueOfFlag</a>
 <a href="chap13.html#X78BE3D727B060301">13.1-22 ResidueOfFlag</a>
 <a href="chap13.html#X815BE6D57D623452">13.1-23 IncidenceGraph</a>
 <a href="chap13.html#X8233EE5D7BC9FD15">13.1-24 Rk2GeoGonality</a>
 <a href="chap13.html#X8573A0DE841CE7B3">13.1-25 Rk2GeoDiameter</a>
 <a href="chap13.html#X8327F4A481DD6DEB">13.1-26 GeometryOfRank2Residue</a>
 <a href="chap13.html#X7B63DAB082839BD3">13.1-27 Rank2Parameters</a>
</div></div>
<div class="ContSect"> <a href="chap13.html#X7967CA67876214A6">13.2 Automorphisms, Correlations and Isomorphisms</a>

<div class="ContSSBlock">
 <a href="chap13.html#X833D384D7B061881">13.2-1 AutGroupIncidenceStructureWithNauty</a>
 <a href="chap13.html#X85260FB17EACF6F5">13.2-2 CorGroupIncidenceStructureWithNauty</a>
 <a href="chap13.html#X82CB6DDB8068AC42">13.2-3 IsIsomorphicIncidenceStructureWithNauty</a>
</div></div>
<div class="ContSect"> <a href="chap13.html#X78932FB48237B18F">13.3 Diagrams</a>

<div class="ContSSBlock">
 <a href="chap13.html#X8239310E79D3DE12">13.3-1 DiagramOfGeometry</a>
 <a href="chap13.html#X7A2783337F87104F">13.3-2 GeometryOfDiagram</a>
 <a href="chap13.html#X808A503579A155BF">13.3-3 DrawDiagram</a>
 <a href="chap13.html#X79F882017FEFE4DE">13.3-4 DrawDiagramWithNeato</a>
</div></div>
</div>

<h3>13 Coset Geometries and Diagrams</h3>

This part of FinInG depends on GRAPE.

<a id="X781B20AC8097AC9F" name="X781B20AC8097AC9F"></a>

<h4>13.1 Coset Geometries</h4>

Suppose we have an incidence geometry Γ (as defined in chapter <a href="chap3.html#X838ACF8A7F100A2B">3</a>), together with a group G of automorphisms of Γ such that G is transitive on the set of chambers of Γ (also defined in chapter <a href="chap3.html#X838ACF8A7F100A2B">3</a>). This implies that G is also transitive on the set of all elements of any chosen type i. If we consider a chamber c1,c2,...,cn such that ci is of type i, we can look at the stabilizer Gi of ci in G. The subgroups Gi are called parabolic subgroups of Γ. For a type i, transitivity of G on the elements of type i gives a correspondence between the cosets of the stabilizer Gi and the elements of type i in Γ. Two elements of Γ are incident if and only if the corresponding cosets have a nonempty intersection.

We now use the above observation to define an incidence structure from a group G together with a set of subgroups G1,G2,...,Gn. The type set is {1,2,...,n}. By definition the elements of type i are the (right) cosets of the subgroup Gi. Two cosets are incident if and only if their intersection is not empty. This is an incidence structure which is not necessarily a geometry (see Chapter <a href="chap3.html#X838ACF8A7F100A2B">3</a> for definitions). In order to check whether a coset incidence structure is indeed a geometry you can use the command <code class="file">IsFlagTransitiveGeometry</code> which (in case it returns <code class="file">true</code>) guarantees that the argument is a geometry.

<a id="X877735BF810DEAD5" name="X877735BF810DEAD5"></a>

<h5>13.1-1 IsCosetGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCosetGeometry</code></td><td class="tdright">( category )</td></tr></table></div>
This category is a subcategory of <code class="code">IsIncidenceGeometry</code>, and contains all coset geometries.

<a id="X86C97C4780C27E6F" name="X86C97C4780C27E6F"></a>

<h5>13.1-2 CosetGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CosetGeometry</code>( <var class="Arg">G</var>, <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the coset incidence structure defined by the list <var class="Arg">l</var> of subgroups of the group <var class="Arg">G</var>.

<var class="Arg">G</var> must be a group and <var class="Arg">l</var> is a list of subgroups of <var class="Arg">G</var>. The subgroups in <var class="Arg">l</var> will be the parabolic subgroups of the coset incidence structure whose rank equals the length of <var class="Arg">l</var>.

<div class="example"><pre>
gap> g:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> g1:=Stabilizer(g,[1,2],OnSets);
Group([ (4,5), (3,5), (1,2)(4,5) ])
gap> g2:=Stabilizer(g,[1,2,3],OnSets);
Group([ (4,5), (2,3), (1,2,3) ])
gap> cg:=CosetGeometry(g,[g1,g2]);
CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )
gap> p:=Random(ElementsOfIncidenceStructure(cg,1));
<element of type 1 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )>
gap> q:=Random(ElementsOfIncidenceStructure(cg,2));
<element of type 2 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )>
gap> IsIncident(p,q);
false
gap> IsIncident(p,p);
true
gap> ParabolicSubgroups(cg);
[ Group([ (4,5), (3,5), (1,2)(4,5) ]), Group([ (4,5), (2,3), (1,2,3) ]) ]
gap> Rank(cg) = Size(last);
true
gap> BorelSubgroup(cg);
Group([ (1,2), (4,5) ])
gap> AmbientGroup(cg);
Sym( [ 1 .. 5 ] )

</pre></div>

<a id="X7A9ED8327C40B445" name="X7A9ED8327C40B445"></a>

<h5>13.1-3 IsIncident</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIncident</code>( <var class="Arg">ele1</var>, <var class="Arg">ele2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if <var class="Arg">ele1</var> and <var class="Arg">ele2</var> are incident

<var class="Arg">ele1</var> and <var class="Arg">ele2</var> must be two elements in the same coset geometry.

<a id="X86CC6815824E71E4" name="X86CC6815824E71E4"></a>

<h5>13.1-4 ParabolicSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ParabolicSubgroups</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the list of parabolic subgroups defining the coset geometry <var class="Arg">cg</var>

<a id="X824D5D558425B4B5" name="X824D5D558425B4B5"></a>

<h5>13.1-5 AmbientGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AmbientGroup</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the group used to define the coset geometry <var class="Arg">cg</var>

<var class="Arg">cg</var> must be a coset geometry.

<a id="X7E5B687E86068196" name="X7E5B687E86068196"></a>

<h5>13.1-6 Borelsubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Borelsubgroup</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the Borel subgroup of de geometry <var class="Arg">cg</var>

The Borel subgroup is equal to the stabilizer of a chamber. It corresponds to the intersection of all parabolic subgroups.

<a id="X82FB0B387E53B073" name="X82FB0B387E53B073"></a>

<h5>13.1-7 RandomElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomElement</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a random element of <var class="Arg">cg</var>

<var class="Arg">cg</var> must be a coset geometry.

<a id="X87E9C0D3799F95BA" name="X87E9C0D3799F95BA"></a>

<h5>13.1-8 RandomFlag</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomFlag</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a random flag of <var class="Arg">cg</var>

<var class="Arg">cg</var> must be a coset geometry.

<a id="X82DA90817A70103A" name="X82DA90817A70103A"></a>

<h5>13.1-9 RandomChamber</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomChamber</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a random chamber of <var class="Arg">cg</var>

<var class="Arg">cg</var> must be a coset geometry.

<div class="example"><pre>
gap> g:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> g1:=Stabilizer(g,[1,2],OnSets);
Group([ (4,5), (3,5), (1,2)(4,5) ])
gap> g2:=Stabilizer(g,[[1,2],[3,4]],OnSetsSets);
Group([ (1,2), (3,4), (1,3)(2,4) ])
gap> cg:=CosetGeometry(g,[g1,g2]);
CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )
gap> RandomElement(cg);
<element of type 1 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )>
gap> Display(last);
RightCoset(Group( [ (4,5), (3,5), (1,2)(4,5) ] ),(1,4,2,5,3))
gap> RandomFlag(cg);
<Flag of coset geometry < CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) ,
[ Group( [ (4,5), (3,5), (1,2)(4,5) ] ),
 Group( [ (1,2), (3,4), (1,3)(2,4) ] ) ] ) >>
gap> flg:=RandomFlag(cg);
<Flag of coset geometry < CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) ,
[ Group( [ (4,5), (3,5), (1,2)(4,5) ] ),
 Group( [ (1,2), (3,4), (1,3)(2,4) ] ) ] ) >>
gap> Type(flg);
[ 1 ]
gap> flg2:=RandomFlag(cg);
<Flag of coset geometry < CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) ,
[ Group( [ (4,5), (3,5), (1,2)(4,5) ] ),
 Group( [ (1,2), (3,4), (1,3)(2,4) ] ) ] ) >>
gap> Type(flg2);
[ 2 ]
gap> IsChamberOfIncidenceStructure(flg2);
false
gap> IsChamberOfIncidenceStructure(flg);
false
gap> Display(flg2);
Flag of coset geometry CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) ,
[ Group( [ (4,5), (3,5), (1,2)(4,5) ] ),
 Group( [ (1,2), (3,4), (1,3)(2,4) ] ) ] ) with elements
[ RightCoset(Group( [ (1,2), (3,4), (1,3)(2,4) ] ),(2,3,5)) ]
gap> cham:=RandomChamber(cg);
<Flag of coset geometry < CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) ,
[ Group( [ (4,5), (3,5), (1,2)(4,5) ] ),
 Group( [ (1,2), (3,4), (1,3)(2,4) ] ) ] ) >>
gap> IsChamberOfIncidenceStructure(cham);
true
gap> ElementsOfFlag(cham);
[ <element of type 1 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )>,
 <element of type 2 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )> ]
gap> IsIncident(last[1],last[2]);
true

</pre></div>

<a id="X7FEFCB907CAAA9B1" name="X7FEFCB907CAAA9B1"></a>

<h5>13.1-10 IsFlagTransitiveGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFlagTransitiveGeometry</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if the group <var class="Arg">G</var> defining <var class="Arg">cg</var> acts flag-transitively.

<var class="Arg">cg</var> must be a coset geometry.

The group G used to define <var class="Arg">cg</var> acts naturally on the elements of <var class="Arg">cg</var> by right translation: a coset G_ig is mapped to G_i(gx) by an element x∈ G. This test can be quite time consuming. You can bind the attribute <code class="code">IsFlagTransitiveGeometry</code> if you are sure the coset geometry is indeed flag-transitive.

<div class="example"><pre>
gap> g:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> g1:=Subgroup(g,[(1,2,3)]);
Group([ (1,2,3) ])
gap> g2:=Subgroup(g,[(1,4)]);
Group([ (1,4) ])
gap> g3:=Subgroup(g,[(1,2,3,4)]);
Group([ (1,2,3,4) ])
gap> cg:=CosetGeometry(g,[g1,g2,g3]);
CosetGeometry( SymmetricGroup( [ 1 .. 4 ] ) )
gap> IsFlagTransitiveGeometry(cg);
false
gap> cg2:=CosetGeometry(g,[g1,g2]);
CosetGeometry( SymmetricGroup( [ 1 .. 4 ] ) )
gap> IsFlagTransitiveGeometry(cg2);
true

</pre></div>

<a id="X7C0E5EDD806CE4C1" name="X7C0E5EDD806CE4C1"></a>

<h5>13.1-11 OnCosetGeometryElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnCosetGeometryElement</code>( <var class="Arg">ele</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the image of the <code class="file">CosetGeometryElement</code> <var class="Arg">ele</var> under the action of <var class="Arg">g</var>

The group element <var class="Arg">g</var> must belong to <code class="file">AmbientGroup(AmbientGeometry(ele))</code>.

<a id="X7D21FB1A7D21FB1A" name="X7D21FB1A7D21FB1A"></a>

<h5><code>13.1-12 \^</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \^</code>( <var class="Arg">ele</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: an element of a coset geometry

This is an operation which returns the image of <var class="Arg">ele</var>, an element of a coset incidence structure, under <var class="Arg">g</var>, an element of <code class="file">AmbientGroup(AmbientGeometry(ele))</code>.

<a id="X7D21FB1A7D21FB1A" name="X7D21FB1A7D21FB1A"></a>

<h5><code>13.1-13 \^</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \^</code>( <var class="Arg">flg</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a flag of a coset geometry

This is an operation which returns the image of <var class="Arg">flg</var>, a flag of a coset incidence structure, under <var class="Arg">g</var>, an element of <code class="file">AmbientGroup(AmbientGeometry(flg))</code>.

<a id="X7F74FD648751B7E6" name="X7F74FD648751B7E6"></a>

<h5>13.1-14 IsFirmGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFirmGeometry</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if <var class="Arg">cg</var> is firm.

An incidence geometry is said to be firm if every non-maximal flag is contained in at least two chambers. <var class="Arg">cg</var> must be a coset geometry.

<a id="X7F8A17357F9FFA00" name="X7F8A17357F9FFA00"></a>

<h5>13.1-15 IsThickGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsThickGeometry</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if <var class="Arg">cg</var> is thick.

An incidence geometry is said to be thick if every non-maximal flag is contained in at least three chambers. <var class="Arg">cg</var> must be a coset geometry.

<a id="X830213C3843A1BA2" name="X830213C3843A1BA2"></a>

<h5>13.1-16 IsThinGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsThinGeometry</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if <var class="Arg">cg</var> is thin.

An incidence geometry is said to be thin if every rank one residue contains exactly 2 elements. This means that every comaximal flag is contained in exactly 2 chambers. <var class="Arg">cg</var> must be a coset geometry.

<div class="example"><pre>
gap> g:=SymmetricGroup(8);;
gap> pabs:=[];;
gap> pabs[1]:=Stabilizer(g,1);; pabs[2]:=Stabilizer(g,2);;
gap> pabs[3]:=Stabilizer(g,3);;
gap> pabs[4]:=Stabilizer(g,[1,2,3,4],OnSets);;
gap> pabs[5]:=Stabilizer(g,[1,2,3,4,5],OnSets);;
gap> pabs[6]:=Stabilizer(g,6);; pabs[7]:=Stabilizer(g,7);;
gap> cg:=CosetGeometry(g,pabs);
CosetGeometry( SymmetricGroup( [ 1 .. 8 ] ) )
gap> IsFirmGeometry(cg);
true
gap> IsThinGeometry(cg);
true
gap> IsThickGeometry(cg);
false
gap> truncation:=CosetGeometry(g,pabs{[1..5]});
CosetGeometry( SymmetricGroup( [ 1 .. 8 ] ) )
gap> IsFirmGeometry(truncation);
true
gap> IsThinGeometry(truncation);
false
gap> IsThickGeometry(truncation);
false
gap> truncation2:=CosetGeometry(g,pabs{[4,5]});
CosetGeometry( SymmetricGroup( [ 1 .. 8 ] ) )
gap> IsFirmGeometry(truncation2);
true
gap> IsThinGeometry(truncation2);
false
gap> IsThickGeometry(truncation2);
true

</pre></div>

<a id="X877F178084D9A589" name="X877F178084D9A589"></a>

<h5>13.1-17 IsConnected</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConnected</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if <var class="Arg">cg</var> is connected.

A geometry is connected if and only if its incidence graph is connected. <var class="Arg">cg</var> must be a coset geometry.

<a id="X83A5107981824C2F" name="X83A5107981824C2F"></a>

<h5>13.1-18 IsResiduallyConnected</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsResiduallyConnected</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true if and only if <var class="Arg">cg</var> is residually connected.

A geometry is residually connected if the incidence graphs of all its residues of rank at least 2 are connected. <var class="Arg">cg</var> must be a coset geometry.

This test is quite time consuming. You can bind the attribute <code class="code">IsResiduallyConnected</code> if you are sure the coset geometry is indeed residually connected.

<div class="example"><pre>
gap> ps:=HyperbolicQuadric(7,2);
Q+(7, 2)
gap> g:=IsometryGroup(ps);;
gap> reps:=RepresentativesOfElements(ps);
[ <a point in Q+(7, 2)>, <a line in Q+(7, 2)>, <a plane in Q+(7, 2)>,
 <a solid in Q+(7, 2)> ]
gap> solids:=Orbit(g,reps[4]);;
gap> ps:=HyperbolicQuadric(7,2);
Q+(7, 2)
gap> g:=IsometryGroup(ps);;
gap> reps:=RepresentativesOfElements(ps);
[ <a point in Q+(7, 2)>, <a line in Q+(7, 2)>, <a plane in Q+(7, 2)>,
 <a solid in Q+(7, 2)> ]
gap> h:=DerivedSubgroup(g);; # to get greek and latin solids
gap> orbs:=FiningOrbits(h,Solids(ps));;
50%..100%..gap> List(orbs, Size);
[ 135, 135 ]
gap> Filtered(orbs[2], s -> ProjectiveDimension(Meet(orbs[1][1],s))=2); # to
[ <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>,
 <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>,
 <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>,
 <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>,
 <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)>, <a solid in Q+(7, 2)> ]
gap> #find a latin incident with the greek which is orbs[1][1]
gap> # Now we have a chamber
gap> goodreps:=[reps[1],reps[2],orbs[1][1],last[1]];
[ <a point in Q+(7, 2)>, <a line in Q+(7, 2)>, <a solid in Q+(7, 2)>,
 <a solid in Q+(7, 2)> ]
gap> pabs:=List(goodreps, r -> FiningStabiliser(h,r));
[ <projective collineation group of size 1290240 with 2 generators>,
 <projective collineation group of size 110592 with 4 generators>,
 <projective collineation group of size 1290240 with 2 generators>,
 <projective collineation group of size 1290240 with 4 generators> ]
gap> cos:=CosetGeometry(h,pabs);
CosetGeometry( Group(
[ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),8,[
 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),8,[[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),8,[[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0
 ],]),IdentityMapping( GF(2) )) ] ) )
gap> IsConnected(cos);
true
gap> IsResiduallyConnected(cos);
true
gap> time;
419960

</pre></div>

<a id="X80E7BC8F8612176D" name="X80E7BC8F8612176D"></a>

<h5>13.1-19 StandardFlagOfCosetGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardFlagOfCosetGeometry</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: standard chamber of <var class="Arg">cg</var>

The standard chamber just consists of all parabolic subgroups (i.e. the trivial cosets of these subgroups). The object returned is a FlagOfIncidenceStructure. <var class="Arg">cg</var> must be a coset geometry.

<a id="X875E89FF7ECD44A6" name="X875E89FF7ECD44A6"></a>

<h5>13.1-20 FlagToStandardFlag</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FlagToStandardFlag</code>( <var class="Arg">cg</var>, <var class="Arg">fl</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: element of the defining group of <var class="Arg">cg</var> which maps <var class="Arg">fl</var> to the standard chamber of <var class="Arg">cg</var>.

<var class="Arg">fl</var> must be a chamber given as a list of cosets of the parabolic subgroups of <var class="Arg">cg</var>.

<div class="example"><pre>
gap> L:=SimpleLieAlgebra("D",8,Rationals);
<Lie algebra of dimension 120 over Rationals>
gap> rs:=RootSystem(L);
<root system of rank 8>
gap> w:=WeylGroup(rs);
<matrix group with 8 generators>
gap> gens:=GeneratorsOfGroup(w);;
gap> pabs:=List(gens, g -> Group(Difference(gens, [g])));
[ <matrix group with 7 generators>, <matrix group with 7 generators>,
 <matrix group with 7 generators>, <matrix group with 7 generators>,
 <matrix group with 7 generators>, <matrix group with 7 generators>,
 <matrix group with 7 generators>, <matrix group with 7 generators> ]
gap> g:=Group(gens);
<matrix group with 8 generators>
gap> cg:=CosetGeometry(g,pabs);;
gap> cham:=RandomChamber(cg);; # Time of last command: 23945 ms
gap> FlagToStandardFlag(cg,cham); # Time of last command: 1720 ms
[ [ 0, 0, 0, 0, 1, -1, 0, 0 ], [ 0, 0, 0, 1, 0, -1, 0, 0 ],
 [ 0, 0, 0, 1, 0, 0, -1, -1 ], [ 1, -1, 0, 1, 0, 0, -1, -1 ],
 [ 0, -1, 0, 1, 0, 0, -1, -1 ], [ 0, -1, 0, 1, 0, 0, 0, -2 ],
 [ 0, -1, 1, 0, 0, 0, 0, -1 ], [ 0, -1, 0, 1, 0, 0, 0, -1 ] ]
gap> cham^last = StandardFlagOfCosetGeometry(cg); # Time of last command:1005 ms
true

</pre></div>

<a id="X7F3A2B47833EA22F" name="X7F3A2B47833EA22F"></a>

<h5>13.1-21 CanonicalResidueOfFlag</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanonicalResidueOfFlag</code>( <var class="Arg">cg</var>, <var class="Arg">fl</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: coset geometry isomorphic to residue of <var class="Arg">fl</var> in <var class="Arg">cg</var>

<var class="Arg">cg</var> must be a coset incidence structure and <var class="Arg">fl</var> must be a flag in that incidence structure. The returned coset incidence structure for a flag {G_i_1g_i_1,G_i_2g_i_2,... , G_i_kg_i_k} is the coset incidence structure defined by the group H:=∩_j=1^kG_i_j and parabolic subgroups G_j∩ H for j not in the type set {i_1,i_2,... ,i_k} of <var class="Arg">fl</var>.

<a id="X78BE3D727B060301" name="X78BE3D727B060301"></a>

<h5>13.1-22 ResidueOfFlag</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResidueOfFlag</code>( <var class="Arg">fl</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the residue of <var class="Arg">fl</var> in <code class="file">AmbientGeometry(fl)</code>.

This is a <code class="file">CosetGeometry</code> method for the <code class="file">ResidueOfFlag</code> operation given in Chapter <a href="chap3.html#X838ACF8A7F100A2B">3</a>. Note that the related operation <code class="file">CanonicalResidueOfFlag</code> takes two arguments.

<div class="example"><pre>
gap> pg:=SymplecticSpace(5,2);
W(5, 2)
gap> pi:=Random(Planes(pg));
<a plane in W(5, 2)>
gap> l:=Random(Lines(pi));
<a line in W(5, 2)>
gap> p:=Random(Points(l));
<a point in W(5, 2)>
gap> g:=CollineationGroup(pg);
PGammaSp(6,2)
gap> g1:=Stabilizer(g,p);
<projective collineation group of size 23040 with 3 generators>
gap> g2:=Stabilizer(g,l);
<projective collineation group of size 4608 with 4 generators>
gap> g3:=Stabilizer(g,pi);
<projective collineation group of size 10752 with 3 generators>
gap> cg:=CosetGeometry(g, [g1,g2,g3]);
CosetGeometry( PGammaSp(6,2) )
gap> RandomFlag(cg); # Time of last command: 10745 ms
<Flag of coset geometry < CosetGeometry( PGammaSp(6,2) ,
[
 Group(
 [ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2)
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )) ] ),
 Group(
 [ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )) ] ),
 Group(
 [ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0
 ],]),IdentityMapping( GF(2) )),
 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2)
 ],]),IdentityMapping( GF(2) )) ] ) ] ) >>
gap> Type(last);
[ ]
gap> ResidueOfFlag(last2);
CosetGeometry( PGammaSp(6,2) )
gap> Rank(last);
3
gap> NrElementsOfIncidenceStructure(last2,1);
63
gap> flg:=RandomFlag(cg);;
gap> can:=CanonicalResidueOfFlag(cg,flg);
CosetGeometry( Group( ... ) )
gap> Type(flg);
[ 1, 2 ]
gap> Rank(can);
1
gap> res:=ResidueOfFlag(flg);
CosetGeometry( Group(
[ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),6,[[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),6,[[ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),6,[[ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),6,[[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,
 1),6,[[ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2)
 ],]),IdentityMapping( GF(2) )) ] ) )
gap> IsIsomorphicIncidenceStructureWithNauty(res,can);
#I Using NiceMonomorphism...
#I Using NiceMonomorphism...
true

</pre></div>

<a id="X815BE6D57D623452" name="X815BE6D57D623452"></a>

<h5>13.1-23 IncidenceGraph</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IncidenceGraph</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: incidence graph of <var class="Arg">cg</var>.

<var class="Arg">cg</var> must be a coset geometry. The graph returned is a GRAPE object. All GRAPE functionality can now be used to analyse <var class="Arg">cg</var> via its incidence graph.

<a id="X8233EE5D7BC9FD15" name="X8233EE5D7BC9FD15"></a>

<h5>13.1-24 Rk2GeoGonality</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rk2GeoGonality</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the gonality (i.e. half the girth) of the incidence graph of <var class="Arg">cg</var>.

<var class="Arg">cg</var> must be a coset geometry of rank 2.

<a id="X8573A0DE841CE7B3" name="X8573A0DE841CE7B3"></a>

<h5>13.1-25 Rk2GeoDiameter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rk2GeoDiameter</code>( <var class="Arg">cg</var>, <var class="Arg">type</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the point (or line) diameter.

<var class="Arg">cg</var> must be a coset geometry of rank 2. <var class="Arg">type</var> must be either <code class="code">1</code> or <code class="code">2</code>. This function computes the point diameter of <var class="Arg">cg</var> when <var class="Arg">type</var> is <code class="code">1</code> and the line diameter when <var class="Arg">type</var> is <code class="code">2</code>.

<a id="X8327F4A481DD6DEB" name="X8327F4A481DD6DEB"></a>

<h5>13.1-26 GeometryOfRank2Residue</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeometryOfRank2Residue</code>( <var class="Arg">resi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the geometry of the <code class="file">Rank2Residue</code> object <var class="Arg">resi</var>.

The rank 2 residues of a geometry are fundamental when dealing with diagrams. Therefore they are kept in an attribute as (a list of) objects of type <code class="file">Rank2Residue</code>. The present operation just extracts the residue as a coset geometry from such a <code class="file">Rank2Residue</code> object.

<a id="X7B63DAB082839BD3" name="X7B63DAB082839BD3"></a>

<h5>13.1-27 Rank2Parameters</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rank2Parameters</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a list of length 3.

<var class="Arg">cg</var> must be a coset geometry of rank 2. This function computes the gonality, point and line diameter of <var class="Arg">cg</var>. These appear as a list in the first entry of the returned list. The second entry contains a list of length 2 with the point order and the total number of points (i.e. elements of type 1) in the geometry. The last entry contains the line order and the number of lines (i.e. elements of type 2).

The following example illustrates <code class="file">Rank2Parameters</code>. It uses AtlasRep to fetch the second Janko group, also known as the Hall-Janko group. Beware that AtlasRep needs special write permissions on some systems. The constructed geometry has gonality 6 and both diameters equal to 8. It is known as the Cohen-Tits near octagon.

<div class="example"><pre>
gap> LoadPackage("atlasrep");
true
gap> j2:=AtlasGroup("J2"); #Uses AtlasRep package
<permutation group of size 604800 with 2 generators>
gap> max3:=AtlasSubgroup(j2,3); #member of 3rd ATLAS class of max. subgps
<permutation group of size 1920 with 2 generators>
gap> max4:=AtlasSubgroup(j2,4); #member of 4th ATLAS class of max. subgps
<permutation group of size 1152 with 2 generators>
gap> conj3:=ConjugacyClassSubgroups(j2,max3);;
gap> g1:=First(conj3, c -> Size(Intersection(c,max4))=384);;
gap> g2:=max4;;
gap> cg:=CosetGeometry(j2,[g1,g2]);;
gap> Rank2Parameters(cg);
[ [ 6, 8, 8 ], [ 2, 315 ], [ 4, 525 ] ]

</pre></div>

<a id="X7967CA67876214A6" name="X7967CA67876214A6"></a>

<h4>13.2 Automorphisms, Correlations and Isomorphisms</h4>

An automorphism of an incidence structure Γ is a permutation of the element set of Γ such that incidence is preserved and types are fixed (i.e. the type of the image of an element coincides with the type of that element). One way to compute the (full) automorphism group of Γ is to compute its incidence graph and then use the available <code class="file">nauty</code> machinery to obtain the group.

<a id="X833D384D7B061881" name="X833D384D7B061881"></a>

<h5>13.2-1 AutGroupIncidenceStructureWithNauty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutGroupIncidenceStructureWithNauty</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: permutation group isomorphic to the full automorphism group of <var class="Arg">cg</var>.

The group is computed with <code class="file">nauty</code>, which is part of GRAPE but has to be compiled on your system before use. The group returned is a permutation group acting on the set <code class="file">[1..Sum(TypesOfElementsOfIncidenceStructure(cg), t -> NrElementsOfIncidenceStructure(cg,t))]</code>, which is exactly the vertex set of <code class="file">IncidenceGraph(cg)</code>. At the moment the action of the automorphism group on <code class="file">cg</code> is not provided but it can be recovered from the knowledge that the vertex set of <code class="file">IncidenceGraph(cg)</code> first contains all elements of type 1 in <code class="file">cg</code>, then all elements of type 2, etc. or, better still, with the GRAPE command <code class="file">VertexNames</code> (see example below).

<a id="X85260FB17EACF6F5" name="X85260FB17EACF6F5"></a>

<h5>13.2-2 CorGroupIncidenceStructureWithNauty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CorGroupIncidenceStructureWithNauty</code>( <var class="Arg">cg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: permutation group isomorphic to the full automorphism group of <var class="Arg">cg</var>.

The group is computed with <code class="file">nauty</code>, which is part of GRAPE but has to be compiled on your system before use. The group returned is a permutation group acting on the set <code class="file">[1..Sum(TypesOfElementsOfIncidenceStructure(cg), t -> NrElementsOfIncidenceStructure(cg,t))]</code>, which is exactly the vertex set of <code class="file">IncidenceGraph(cg)</code>. At the moment the action of the automorphism group on <code class="file">cg</code> is not provided but it can be recovered from the knowledge that the vertex set of <code class="file">IncidenceGraph(cg)</code> first contains all elements of type 1 in <code class="file">cg</code>, then all elements of type 2, etc. or with the GRAPE command <code class="file">VertexNames</code>.

<div class="example"><pre>
gap> g := PSL(2,11);;
gap> g1 := Group([ (1,2,3)(4,8,12)(5,10,9)(6,11,7), 
> (1,2)(3,4)(5,12)(6,11)(7,10)(8,9) ]);;
gap> g2 := Group([ (1,2,7)(3,9,4)(5,11,10)(6,8,12), 
> (1,2)(3,4)(5,12)(6,11)(7,10)(8,9) ]);;
gap> g3 := Group([ (1,2,11)(3,8,7)(4,9,5)(6,10,12), 
> (1,2)(3,12)(4,11)(5,10)(6,9)(7,8) ]);;
gap> g4 := Group([ (1,2,11)(3,8,7)(4,9,5)(6,10,12), 
> (1,2)(3,10)(4,9)(5,8)(6,7)(11,12) ]);;
gap> cg:=CosetGeometry(g,[g1,g2,g3,g4]);
CosetGeometry( Group( [ ( 3,11, 9, 7, 5)( 4,12,10, 8, 6),
 ( 1, 2, 8)( 3, 7, 9)( 4,10, 5)( 6,12,11) ] ) )
gap> aut:=AutGroupIncidenceStructureWithNauty(cg);
<permutation group with 4 generators>
gap> StructureDescription(aut);
"PSL(2,11)"
gap> cor:=CorGroupIncidenceStructureWithNauty(cg);
<permutation group with 5 generators>
gap> StructureDescription(cor);
"C2 x PSL(2,11)"
gap> incgrph:=IncidenceGraph(cg);;
gap> names:=VertexNames(incgrph);;
gap> g:=Random(aut);
(1,9,7,6,2,3,5,11,4,8,10)(12,13,15,17,14,19,22,16,18,21,20)(23,28,33,25,29,31,
32,26,27,24,30)(34,44,38,41,42,35,43,39,40,36,37)
gap> e:=RandomElement(cg);
<element of type 3 of CosetGeometry( Group(
[ ( 3,11, 9, 7, 5)( 4,12,10, 8, 6), ( 1, 2, 8)( 3, 7, 9)( 4,10, 5)( 6,12,11)
] ) )>
gap> pos:=Position(names, e);
26
gap> names[pos^g];
<element of type 3 of CosetGeometry( Group(
[ ( 3,11, 9, 7, 5)( 4,12,10, 8, 6), ( 1, 2, 8)( 3, 7, 9)( 4,10, 5)( 6,12,11)
] ) )>

</pre></div>

<a id="X82CB6DDB8068AC42" name="X82CB6DDB8068AC42"></a>

<h5>13.2-3 IsIsomorphicIncidenceStructureWithNauty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIsomorphicIncidenceStructureWithNauty</code>( <var class="Arg">cg1</var>, <var class="Arg">cg2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true iff cg1 and cg2 are isomorphic.

We use <code class="file">nauty</code>, which is part of GRAPE but has to be compiled on your system before use. Isomorphism is tested (with <code class="file">nauty</code>) after converting the coset geometries <var class="Arg">cg1</var> and <var class="Arg">cg2</var> to coloured graphs.

<div class="example"><pre>
gap> g:=SymmetricGroup(4); g1:=Subgroup(g,[(1,2,3)]);
Sym( [ 1 .. 4 ] )
Group([ (1,2,3) ])
gap> g2:=Subgroup(g,[(1,4)]); g3:=Subgroup(g,[(1,2,3,4)]);
Group([ (1,4) ])
Group([ (1,2,3,4) ])
gap> cg:=CosetGeometry(g,[g1,g2,g3]);
CosetGeometry( SymmetricGroup( [ 1 .. 4 ] ) )
gap> IsFlagTransitiveGeometry(cg);
false
gap> aut:=AutGroupIncidenceStructureWithNauty(cg);
<permutation group with 4 generators>
gap> Size(aut);
48
gap> Size(g);
24
gap> newg1:=Stabilizer(aut, 1);
Group([ (5,7)(6,8)(10,15)(11,12)(13,16)(14,18)(17,19)(21,25)(23,26), (3,6)
(4,5)(9,18)(10,16)(12,20)(13,17)(15,19)(21,22)(24,26) ])
gap> newg2:=Stabilizer(aut, NrElementsOfIncidenceStructure(cg,1) + 1);
Group([ (5,7)(6,8)(10,15)(11,12)(13,16)(14,18)(17,19)(21,25)(23,26), (1,3)
(2,4)(5,6)(7,8)(10,11)(12,15)(13,16)(14,17)(18,19)(21,25)(22,24)(23,26) ])
gap> newg3:=Stabilizer(aut, NrElementsOfIncidenceStructure(cg,1) +
> NrElementsOfIncidenceStructure(cg,2) + 1);
Group([ (1,3)(2,4)(5,8)(6,7)(10,12)(11,15)(14,19)(17,18)(22,24), (3,8)(4,7)
(9,14)(10,17)(11,20)(13,15)(16,19)(22,25)(23,24) ])
gap> newcg:=CosetGeometry(aut, [newg1, newg2, newg3]);
CosetGeometry( Group(
[ ( 5, 7)( 6, 8)(10,15)(11,12)(13,16)(14,18)(17,19)(21,25)(23,26),
 ( 3, 6)( 4, 5)( 9,18)(10,16)(12,20)(13,17)(15,19)(21,22)(24,26),
 ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,20)(10,17)(11,14)(12,18)(13,16)(15,19)
 (21,26)(22,24)(23,25), ( 1, 3)( 2, 4)( 5, 6)( 7, 8)(10,11)(12,15)(13,16)
 (14,17)(18,19)(21,25)(22,24)(23,26) ] ) )
gap> IsFlagTransitiveGeometry(newcg);
true
gap> IsIsomorphicIncidenceStructureWithNauty(cg, newcg);
true

</pre></div>

<a id="X78932FB48237B18F" name="X78932FB48237B18F"></a>

<h4>13.3 Diagrams</h4>

The diagram of a flag-transitive incidence geometry is a schematic description of the structure of the geometry. It is based on the collection of rank 2 residues of the geometry.

Technically, the diagram is added to a <code class="file">CosetGeometry</code> object as a mutable attribute. Also the list of rank 2 residues of the geometry is added as an attribute once these have been computed. This is done with the operations <code class="file">Rank2Residues</code> (to add the attribute) and <code class="file">MakeRank2Residue</code> (to actually compute the residues). These operations are not for everyday use and hence remain undocumented.

Since the geometry is flag-transitive, all chambers are equivalent. Let's fix a chamber C = { c1,c2,...,cn}, with ci of type i. For each subset {i,j} of size two in {1,2,...n} we take the residue of the flag C \ { ci, cj}. Flag transitivity ensures that all residues of type {i,j} are isomorphic to each other. For each such residue, the structure is described by some parameters: the gonality and the point and line diameters. For each type i, we also define the i-order to be one less than the number of elements of type i in the residue of a(ny) flag of type I∖{i}. All this information is depicted in a diagram which is basically a labelled graph with vertex set I and edges whenever the point diameter, the line diameter and the gonality are all greater than 2.

<a id="X8239310E79D3DE12" name="X8239310E79D3DE12"></a>

<h5>13.3-1 DiagramOfGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DiagramOfGeometry</code>( <var class="Arg">Gamma</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the diagram of the geometry <var class="Arg">Gamma</var>

<var class="Arg">Gamma</var> must be a flag-transitive coset geometry.

The flag-transitivity is not tested by this operation because such a test can be time consuming. The command <code class="code">IsFlagTransitiveGeometry</code> can be used to check flag-transitivity if needed.

<a id="X7A2783337F87104F" name="X7A2783337F87104F"></a>

<h5>13.3-2 GeometryOfDiagram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeometryOfDiagram</code>( <var class="Arg">diag</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the geometry of which <var class="Arg">diag</var> is the diagram

<var class="Arg">diag</var> must be a diagram object.

<a id="X808A503579A155BF" name="X808A503579A155BF"></a>

<h5>13.3-3 DrawDiagram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DrawDiagram</code>( <var class="Arg">diag</var>, <var class="Arg">filename</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DrawDiagram</code>( <var class="Arg">diag</var>, <var class="Arg">filename</var>, <var class="Arg">vertexverbosity</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DrawDiagram</code>( <var class="Arg">diag</var>, <var class="Arg">filename</var>, <var class="Arg">vertexverbosity</var>, <var class="Arg">edgeverbosity</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: does not return anything but writes a file <var class="Arg">filename</var>.ps

<var class="Arg">diag</var> must be a diagram. Writes a file <var class="Arg">filename</var>.ps in the current directory with a pictorial version of the diagram. This command uses the graphviz package which is available from http://www.graphviz.org.

In case graphviz is not available on your system, you will get an friendly error message and a file <var class="Arg">filename</var>.dot will be written. You can then compile this file later or ask a friend to help you. By default the diagram provides for each type i the i-order and the number of elements of type i. This behaviour can be changed by providing a <var class="Arg">vertexverbosity</var> level. A value <var class="Arg">2</var> results in no label under the vertices and a value <var class="Arg">1</var> gives only the i-order. Any other positive integer value yields the default behaviour. The default labels for the edges of the diagram use the standard convention that a [g,dp,dl]-gon with all three parameters equal is labelled only with the number g. Putting <var class="Arg">edgeverbosity</var> equal to 2 puts no labels at all. This yields the so called ``basic diagram'' of the geometry. With <var class="Arg">edgeverbosity</var> equal to any integer greater than 2 all labels contain girth and both diameters.

We illustrate the diagram feature with Neumaier's A8-geometry. The affine space of dimension 3 over the field with two elements is denoted by AG(3,2). If we fix a plane Π in PG(3,2), the structure induced on the 8 points not in Π by the lines and planes of PG(3,2) is isomorphic to AG(3,2). Since every two points of AG(3,2) define a line, the collinearity graph of AG(3,2) (that is the graph whose vertices are the points of AG(3,2) and in which two vertices are adjacent whenever they are collinear) is the complete graph K8 on 8 vertices. Given two copies of the complete graph on 8 vertices, one can label the vertices of each of them with the numbers from 1 to 8. These labelings are always equivalent when the two copies are seen as graphs, but not if they are understood as models of the affine space. The reason is that an affine space has parallel lines and to be affinely equivalent, the labelings must be such that edges which were parallel in the first labeling remain parallel in the second labeling. In fact there are 15 affinely nonequivalent ways to label the vertices of K8. The affine space has 14 planes of 4 points and there are 70 subsets of 4 elements in the vertex set of K8. Each time we label K8, there are 14 of the 70 sets of 4 elements which become planes of AG(3,2). The remaining 4-subsets will be called nonplanes for that labeling. A well-known rank 4 geometry discovered by Neumaier in 1984 can be described using these concepts. This geometry is quite important since its residue of cotype 1 is the famous A7-geometry which is known to be the only flag-transitive locally classical C_3-geometry which is not a polar space (see Aschbacher1984 for details). The Neumaier geometry can be constructed as follows. The elements of types 1 and 2 are the vertices and edges of the complete graph K8, the elements of type 3 are the 4-subsets of the vertex set of K8 and the elements of type 4 are the 15 nonequivalent labelings of K8. Incidences are mostly the natural ones. A 4-subset is incident with a labeling of K8 if it is the set of points of a nonplane in the model of AG(3,2) defined by the labeling.

<div class="example"><pre>
Alt( [ 1 .. 8 ] )
gap> pabs:= [
> Group([ (2,4,6), (1,3,2)(4,8)(6,7) ]), 
> Group([ (1,6,7,8,4), (2,5)(3,4) ]),
> Group([ (3,6)(7,8), (2,4,5), (1,5)(2,4), (2,4)(6,7), (6,8,7), 
> (1,2)(4,5), (3,7)(6,8) ]),
> Group([ (1,7,8,4)(2,5,3,6), (1,3)(2,6)(4,8)(5,7), (1,5)(2,4)(3,7)(6,8),
> (1,8)(2,7)(3,4)(5,6), (1,3)(2,6)(4,7)(5,8) ]) ];
[ Group([ (2,4,6), (1,3,2)(4,8)(6,7) ]), Group([ (1,6,7,8,4), (2,5)(3,4) ]),
 Group([ (3,6)(7,8), (2,4,5), (1,5)(2,4), (2,4)(6,7), (6,8,7), (1,2)
 (4,5), (3,7)(6,8) ]), Group([ (1,7,8,4)(2,5,3,6), (1,3)(2,6)(4,8)
 (5,7), (1,5)(2,4)(3,7)(6,8), (1,8)(2,7)(3,4)(5,6), (1,3)(2,6)(4,7)(5,8) ]) ]
gap> cg:=CosetGeometry(g,pabs);
CosetGeometry( AlternatingGroup( [ 1 .. 8 ] ) )
gap> diag:=DiagramOfGeometry(cg);
< Diagram of CosetGeometry( AlternatingGroup( [ 1 .. 8 ] ) ,
[ Group( [ (2,4,6), (1,3,2)(4,8)(6,7) ] ),
 Group( [ (1,6,7,8,4), (2,5)(3,4) ] ),
 Group( [ (3,6)(7,8), (2,4,5), (1,5)(2,4), (2,4)(6,7), (6,8,7), (1,2)(4,5),
 (3,7)(6,8) ] ),
 Group( [ (1,7,8,4)(2,5,3,6), (1,3)(2,6)(4,8)(5,7), (1,5)(2,4)(3,7)(6,8),
 (1,8)(2,7)(3,4)(5,6), (1,3)(2,6)(4,7)(5,8) ] ) ] ) >
gap> DrawDiagram(diag, "neuma8");
gap> #Exec("gv neuma8.ps");
gap> point:=Random(ElementsOfIncidenceStructure(cg,1));
<element of type 1 of CosetGeometry( AlternatingGroup( [ 1 .. 8 ] ) )>
gap> residue:=ResidueOfFlag(FlagOfIncidenceStructure(cg,[point]));
CosetGeometry( Group( [ (3,5,7), (1,7)(2,4,3)(5,8) ] ) )
gap> diagc3:=DiagramOfGeometry(residue);
< Diagram of CosetGeometry( Group( [ (3,5,7), (1,7)(2,4,3)(5,8) ] ) ,
[ Group( [ (4,5,8), (1,4,5), (1,7,8), (1,8,4,2,7) ] ),
 Group( [ (1,8)(4,7), (2,5,3), (1,7)(2,3), (1,7,8), (1,4)(7,8) ] ),
 Group( [ (1,5,4,3)(7,8), (2,4)(5,8) ] ) ] ) >
gap> DrawDiagram(diagc3, "a7geo");
gap> #Exec("gv a7geo.ps");

</pre></div>

The produced diagrams are included here: Neumaier's A8

<img src="./graphics/neuma8.jpg"></img>

The A7 geometry:

<img src="./graphics/a7geo.jpg"></img>

On a UNIX system we can start an external viewer (``gv'' or ghostview in this case) from within GAP with the <code class="file">Exec</code> command.

<a id="X79F882017FEFE4DE" name="X79F882017FEFE4DE"></a>

<h5>13.3-4 DrawDiagramWithNeato</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DrawDiagramWithNeato</code>( <var class="Arg">diag</var>, <var class="Arg">filename</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: does not return anything but writes a file <var class="Arg">filename</var>.ps

<var class="Arg">diag</var> must be a diagram. Writes a file <var class="Arg">filename</var>.ps in the current directory with a pictorial version of the diagram.

This command uses a "spring tension" algorithm to draw the diagram <var class="Arg">diag</var> with straight edges. For some diagrams this looks better than the result of <code class="file">DrawDiagram</code>. However this algorithm does not print the vertex labels.

This command uses the graphviz package which is available from http://www.graphviz.org. In case graphviz is not available on your system, you will get an friendly error message and a file <var class="Arg">filename</var>.dot will be written. You can then compile this file later or ask a friend to help you. An E_6 geometry for comparison: on the left hand side we have the output of DrawDiagram and on the right hand side we see the result of DrawDiagramWithNeato

<img src="./graphics/E6.jpg"></img> <img src="./graphics/E6bis.jpg"></img>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap12.html">[Previous Chapter]</a> <a href="chap14.html">[Next Chapter]</a> </div>

<div class="chlinkbot">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapC.html">C</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

<hr />
generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a>
</body>
</html>

Messung V0.5 in Prozent

¤ Dauer der Verarbeitung: 0.19 Sekunden (vorverarbeitet am 2026-04-25) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.