Quelle chap4.html Sprache: HTML

products/sources/formale Sprachen/GAP/pkg/ugaly/doc/chap4.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 (UGALY) - Chapter 4: Examples</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="chap4" 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="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="chap3.html">[Previous Chapter]</a> <a href="chap5.html">[Next Chapter]</a> </div>

<a href="chap4_mj.html">[MathJax on]</a>
<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap4.html#X7A489A5D79DA9E5C">4 Examples</a>
<div class="ContSect"> <a href="chap4.html#X7A91225C7FCEBA7D">4.1 Discrete groups</a>

<div class="ContSSBlock">
 <a href="chap4.html#X78CD1F4D86BF2A1B">4.1-1 LocalActionElement</a>

 <a href="chap4.html#X864633EB84B6C73E">4.1-2 LocalActionGamma</a>

 <a href="chap4.html#X8366310185354A2D">4.1-3 LocalActionDelta</a>

</div></div>
<div class="ContSect"> <a href="chap4.html#X7F731157843279EC">4.2 Maximal extensions</a>

<div class="ContSSBlock">
 <a href="chap4.html#X8348A9F887317289">4.2-1 LocalActionPhi</a>

</div></div>
<div class="ContSect"> <a href="chap4.html#X80EDDA597A66421E">4.3 Normal subgroups and partitions</a>

<div class="ContSSBlock">
 <a href="chap4.html#X8348A9F887317289">4.3-1 LocalActionPhi</a>

</div></div>
<div class="ContSect"> <a href="chap4.html#X784DFD007FE78DE9">4.4 Abelian quotients</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7F4CC068860BA22F">4.4-1 SignHomomorphism</a>
 <a href="chap4.html#X7D8D1FC07BC90971">4.4-2 AbelianizationHomomorphism</a>
 <a href="chap4.html#X83A7A23D875BFAA2">4.4-3 SpheresProduct</a>
 <a href="chap4.html#X79341499795BF8D9">4.4-4 LocalActionPi</a>
</div></div>
<div class="ContSect"> <a href="chap4.html#X87FE512E7DB7346C">4.5 Semidirect products</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7F425DFC8760388F">4.5-1 CompatibleKernels</a>

 <a href="chap4.html#X83920A2C7CC46AC9">4.5-2 LocalActionSigma</a>

</div></div>
<div class="ContSect"> <a href="chap4.html#X7A8A46F4856570BE">4.6 PGL₂ over the p-adic numbers</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7E80A4A27E8D846C">4.6-1 LocalActionPGL2Qp</a>
 <a href="chap4.html#X87D6A7B679C668A8">4.6-2 LocalActionPSL2Qp</a>
</div></div>
</div>

<h3>4 Examples</h3>

Several classes of examples of subgroups of \mathrm{Aut}(B_{d,k}) that satisfy (C) and or (D) are constructed in <a href="chapBib.html#biBTor20">[Tor20]</a> and implemented in this section. For a given permutation group F\le S_{d}, there are always the three local actions \Gamma(F), \Delta(F) and \Phi(F) on \mathrm{Aut}(B_{d,2}) that project onto F. For some F, these are all distinct and yield all universal groups that have F as their 1-local action, see <a href="chapBib.html#biBTor20">[Tor20, Theorem 3.32]</a>. More examples arise in particular when either point stabilizers in F are not simple, F preserves a partition, or F is not perfect. This section also includes functions to provide the k-local actions of the groups \mathrm{PGL}(2,\mathbb{Q}_{p}) and \mathrm{PSL}(2,\mathbb{Q}_{p}) acting on T_{p+1}.

<a id="X7A91225C7FCEBA7D" name="X7A91225C7FCEBA7D"></a>

<h4>4.1 Discrete groups</h4>

Here, we implement the local actions \Gamma(F),\Delta(F)\le\mathrm{Aut}(B_{d,2}), both of which satisfy both (C) and (D), see <a href="chapBib.html#biBTor20">[Tor20, Section 3.4.1]</a>.

<a id="X78CD1F4D86BF2A1B" name="X78CD1F4D86BF2A1B"></a>

<h5>4.1-1 LocalActionElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionElement</code>( <var class="Arg">d</var>, <var class="Arg">a</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">‣ LocalActionElement</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">a</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">‣ LocalActionElement</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">s</var>, <var class="Arg">addr</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">‣ LocalActionElement</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">a</var></dt>
<dd>Returns: the automorphism \gamma(<var class="Arg">a</var>)=(<var class="Arg">a</var>,(<var class="Arg">a</var>)_{\omega\in\Omega})\in\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3} and a permutation <var class="Arg">a</var> \in S_d.

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">a</var></dt>
<dd>Returns: the automorphism \gamma^{l}(<var class="Arg">a</var>)\in\mathrm{Aut}(B_{d,l}) all of whose 1-local actions are given by <var class="Arg">a</var>.

The arguments of this method are a radius <var class="Arg">l</var> \in\mathbb{N}, a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3} and a permutation <var class="Arg">a</var> \in S_d.

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">s</var>, <var class="Arg">addr</var></dt>
<dd>Returns: the automorphism of B_{d,l} whose 1-local actions are given by <var class="Arg">s</var> at vertices whose address has <var class="Arg">addr</var> as a prefix and are trivial elsewhere.

The arguments of this method are a radius <var class="Arg">l</var> \in\mathbb{N}, a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a permutation <var class="Arg">s</var> \in S_d and an address <var class="Arg">addr</var> of a vertex in B_{d,l} whose last entry is fixed by <var class="Arg">s</var>.

</dd>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">z</var></dt>
<dd>Returns: the automorphism \gamma_{z}(<var class="Arg">aut</var>)=(<var class="Arg">aut</var>,(<var class="Arg">z</var>(<var class="Arg">aut</var>,\omega))_{\omega\in\Omega})\in\mathrm{Aut}(B_{d,k+1}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a radius <var class="Arg">k</var> \in\mathbb{N}, an automorphism <var class="Arg">aut</var> of B_{d,k}, and an involutive compatibility cocycle <var class="Arg">z</var> of a subgroup of \mathrm{Aut}(B_{d,k}) that contains <var class="Arg">aut</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5.html#X80ADE0E379590053">5.3-1</a>)).

</dd>
</dl>

<div class="example"><pre>
gap> LocalActionElement(3,(1,2));
(1,3)(2,4)(5,6)
</pre></div>

<div class="example"><pre>
gap> LocalActionElement(2,3,(1,2));
(1,3)(2,4)(5,6)
gap> LocalActionElement(3,3,(1,2));
(1,5)(2,6)(3,8)(4,7)(9,11)(10,12)
</pre></div>

<div class="example"><pre>
gap> LocalActionElement(3,3,(1,2),[1,3]);
(3,4)
gap> LocalActionElement(3,3,(1,2),[]);
(1,5)(2,6)(3,8)(4,7)(9,11)(10,12)
</pre></div>

<div class="example"><pre>
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> z1:=AllInvolutiveCompatibilityCocycles(S3)[1];;
gap> LocalActionElement(3,1,(1,2),z1);
(1,4)(2,3)(5,6)
gap> z3:=AllInvolutiveCompatibilityCocycles(S3)[3];;
gap> LocalActionElement(3,1,(1,2),z3);
(1,3)(2,4)(5,6)
</pre></div>

<a id="X864633EB84B6C73E" name="X864633EB84B6C73E"></a>

<h5>4.1-2 LocalActionGamma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionGamma</code>( <var class="Arg">d</var>, <var class="Arg">F</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">‣ LocalActionGamma</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">F</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">‣ LocalActionGamma</code>( <var class="Arg">F</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the local action \Gamma(<var class="Arg">F</var>)=\{(a,(a)_{\omega})\mid a\in F\}\le\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, and a subgroup <var class="Arg">F</var> of S_{d}.

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the group \Gamma^{l}(<var class="Arg">F</var>)\le\mathrm{Aut}(B_{d,l}).

The arguments of this method are a radius <var class="Arg">l</var> \in\mathbb{N}, a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, and a subgroup <var class="Arg">F</var> of S_d.

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">z</var></dt>
<dd>Returns: the group \Gamma_{z}(<var class="Arg">F</var>)=\{(a,(<var class="Arg">z</var>(a,\omega))_{\omega\in\Omega})\mid a\in<var class="Arg">F</var>\}\le\mathrm{Aut}(B_{d,k+1}).

The arguments of this method are a local action <var class="Arg">F</var> \le\mathrm{Aut}(B_{d,k}) and an involutive compatibility cocycle <var class="Arg">z</var> of <var class="Arg">F</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5.html#X80ADE0E379590053">5.3-1</a>)).

</dd>
</dl>

<div class="example"><pre>
gap> F:=TransitiveGroup(4,3);;
gap> LocalActionGamma(4,F);
Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,8)(2,7)(3,9)(4,5)(10,12) ])
</pre></div>

<div class="example"><pre>
gap> LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> LocalActionGamma(2,3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> LocalActionGamma(3,3,SymmetricGroup(3));
Group([ (1,8,10)(2,7,9)(3,5,12)(4,6,11), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12) ])
</pre></div>

<div class="example"><pre>
gap> F:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(F);;
gap> H:=LocalActionPi(2,3,F,rho,[1]);;
gap> z:=InvolutiveCompatibilityCocycle(H);;
gap> g:=LocalActionGamma(H,z);;
gap> [NrMovedPoints(g),TransitiveIdentification(g)];
[ 12, 8 ]
</pre></div>

<a id="X8366310185354A2D" name="X8366310185354A2D"></a>

<h5>4.1-3 LocalActionDelta</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionDelta</code>( <var class="Arg">d</var>, <var class="Arg">F</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">‣ LocalActionDelta</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the group \Delta(<var class="Arg">F</var>)\le\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, and a transitive subgroup <var class="Arg">F</var> of S_{d}.

</dd>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">C</var></dt>
<dd>Returns: the group \Delta(<var class="Arg">F</var>,<var class="Arg">C</var>)\le\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a transitive subgroup <var class="Arg">F</var> of S_d, and a central subgroup <var class="Arg">C</var> of the stabilizer <var class="Arg">F</var>_{1} of 1 in <var class="Arg">F</var>.

</dd>
</dl>

<div class="example"><pre>
gap> F:=SymmetricGroup(3);;
gap> D:=LocalActionDelta(3,F);
Group([ (1,3,6)(2,4,5), (1,3)(2,4), (1,2)(3,4)(5,6) ])
gap> F1:=Stabilizer(F,1);;
gap> D1:=LocalActionDelta(3,F,F1);
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6), (1,2)(3,4)(5,6) ])
gap> D=D1;
false
gap> G:=AutBall(3,2);;
gap> D^G=D1^G;
true
</pre></div>

<div class="example"><pre>
gap> F:=PrimitiveGroup(5,3);
AGL(1, 5)
gap> F1:=Stabilizer(F,1);
Group([ (2,3,4,5) ])
gap> C:=Group((2,4)(3,5));
Group([ (2,4)(3,5) ])
gap> Index(F1,C);
2
gap> Index(LocalActionDelta(5,F,F1),LocalActionDelta(5,F,C));
2
</pre></div>

<a id="X7F731157843279EC" name="X7F731157843279EC"></a>

<h4>4.2 Maximal extensions</h4>

For any F\le\mathrm{Aut}(B_{d,k}) that satisfies (C), the group \Phi(F)\le\mathrm{Aut}(B_{d,k+1}) is the maximal extension of F that satisfies (C) as well. It stems from the action of \mathrm{U}_{k}(F) on balls of radius k+1 in T_{d}.

<a id="X8348A9F887317289" name="X8348A9F887317289"></a>

<h5>4.2-1 LocalActionPhi</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">F</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">‣ LocalActionPhi</code>( <var class="Arg">l</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the argument <var class="Arg">F</var></dt>
<dd>Returns: the group \Phi_{k}(<var class="Arg">F</var>)=\{(a,(a_{\omega})_{\omega})\mid a\in <var class="Arg">F</var>,\ \forall \omega\in\Omega:\ a_{\omega}\in C_{F}(a,\omega)\}\le\mathrm{Aut}(B_{d,k+1}).

The argument of this method is a local action <var class="Arg">F</var> \le\mathrm{Aut}(B_{d,k}).

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">F</var></dt>
<dd>Returns: the group \Phi^{l}(<var class="Arg">F</var>)=\Phi_{l-1}\circ\cdots\circ\Phi_{k+1}\circ\Phi_{k}(<var class="Arg">F</var>)\le\mathrm{Aut}(B_{d,l}).

The arguments of this method are a radius <var class="Arg">l</var> \in\mathbb{N} and a local action <var class="Arg">F</var> \le\mathrm{Aut}(B_{d,k}).

</dd>
</dl>

<div class="example"><pre>
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> LocalActionPhi(S3);
Group([ (), (1,4,5)(2,3,6), (1,3)(2,4)(5,6), (1,2), (3,4), (5,6) ])
gap> last=AutBall(3,2);
true
gap> A3:=LocalAction(3,1,AlternatingGroup(3));;
gap> LocalActionPhi(A3);
Group([ (), (1,4,5)(2,3,6) ])
gap> last=LocalActionGamma(3,AlternatingGroup(3));
true
</pre></div>

<div class="example"><pre>
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> groups:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3);
[ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]),
 Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]),
 Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]),
 Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]),
 Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ]
gap> for G in groups do Print(Size(G),",",Size(LocalActionPhi(G)),"\n"); od;
6,6
12,12
24,192
24,192
48,3072
</pre></div>

<div class="example"><pre>
gap> LocalActionPhi(3,LocalAction(4,1,SymmetricGroup(4)));
<permutation group with 34 generators>
gap> last=AutBall(4,3);
true
</pre></div>

<div class="example"><pre>
gap> rho:=SignHomomorphism(SymmetricGroup(3));;
gap> F:=LocalActionPi(2,3,SymmetricGroup(3),rho,[1]);; Size(F);
24
gap> P:=LocalActionPhi(4,F);; Size(P);
12288
gap> IsSubgroup(AutBall(3,4),P);
true
gap> SatisfiesC(P);
true
</pre></div>

<a id="X80EDDA597A66421E" name="X80EDDA597A66421E"></a>

<h4>4.3 Normal subgroups and partitions</h4>

When point stabilizers in F\le S_{d} are not simple, or F preserves a partition, more universal groups can be constructed as follows.

<a id="X8348A9F887317289" name="X8348A9F887317289"></a>

<h5>4.3-1 LocalActionPhi</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">N</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">‣ LocalActionPhi</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">P</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">‣ LocalActionPhi</code>( <var class="Arg">F</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">N</var></dt>
<dd>Returns: the group \Phi(<var class="Arg">F</var>,<var class="Arg">N</var>)\le\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a transitive permutation group <var class="Arg">F</var> \le S_{d} and a normal subgroup <var class="Arg">N</var> of the stabilizer <var class="Arg">F</var>_{1} of 1 in <var class="Arg">F</var>.

</dd>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">P</var></dt>
<dd>Returns: the group \Phi(<var class="Arg">F</var>,<var class="Arg">P</var>)=\{(a,(a_{\omega})_{\omega})\mid a\in <var class="Arg">F</var>,\ a_{\omega}\in C_{F}(a,\omega) constant w.r.t. <var class="Arg">P</var>\}\le\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3} and a permutation group <var class="Arg">F</var> \le S_{d} and a partition <var class="Arg">P</var> of <code class="code">[1..d]</code> preserved by <var class="Arg">F</var>.

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">P</var></dt>
<dd>Returns: the group \Phi_{k}(<var class="Arg">F</var>,<var class="Arg">P</var>)=\{(\alpha,(\alpha_{\omega})_{\omega})\mid \alpha\in <var class="Arg">F</var>,\ \alpha_{\omega}\in C_{F}(\alpha,\omega) constant w.r.t. <var class="Arg">P</var>\}\le\mathrm{Aut}(B_{d,k+1}).

The arguments of this method are a local action <var class="Arg">F</var> \le\mathrm{Aut}(B_{d,k}) and a partition <var class="Arg">P</var> of <code class="code">[1..d]</code> preserverd by \pi<var class="Arg">F</var> \le S_{d}. This method assumes that all compatibility sets with respect to a partition element are non-empty and that all compatibility sets of the identity with respect to a partition element are non-trivial.

</dd>
</dl>

<div class="example"><pre>
gap> F:=SymmetricGroup(4);;
gap> F1:=Stabilizer(F,1);
Sym( [ 2 .. 4 ] )
gap> grps:=NormalSubgroups(F1);
[ Sym( [ 2 .. 4 ] ), Alt( [ 2 .. 4 ] ), Group(()) ]
gap> N:=grps[2];
Alt( [ 2 .. 4 ] )
gap> LocalActionPhi(4,F,N);
Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,4)(2,5)(3,6)(7,8)(10,11), (1,2,3) ])
gap> Index(F1,N);
2
gap> Index(LocalActionPhi(4,F,F1),LocalActionPhi(4,F,N));
16
</pre></div>

<div class="example"><pre>
gap> F:=TransitiveGroup(4,3);
D(4)
gap> P:=Blocks(F,[1..4]);
[ [ 1, 3 ], [ 2, 4 ] ]
gap> G:=LocalActionPhi(4,F,P);
Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,8)(2,7)(3,9)(4,5)(10,12), (1,3)
 (8,9), (4,5)(10,12) ])
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> aut:=Random(mt,G);
(1,3)(4,12)(5,10)(6,11)(8,9)
gap> LocalAction(1,4,2,aut,[1]); LocalAction(1,4,2,aut,[3]);
(2,4)
(2,4)
gap> LocalAction(1,4,2,aut,[2]); LocalAction(1,4,2,aut,[4]);
(1,3)(2,4)
(1,3)(2,4)
</pre></div>

<div class="example"><pre>
gap> H:=TransitiveGroup(4,3);
D(4)
gap> P:=Blocks(H,[1..4]);
[ [ 1, 3 ], [ 2, 4 ] ]
gap> F:=LocalActionPhi(4,H,P);;
gap> G:=LocalActionPhi(F,P);;
gap> SatisfiesC(G);
true
</pre></div>

<a id="X784DFD007FE78DE9" name="X784DFD007FE78DE9"></a>

<h4>4.4 Abelian quotients</h4>

When a permutation group F\le S_{d} is not perfect, i.e. it admits an abelian quotient \rho:F\twoheadrightarrow A, more universal groups can be constructed by imposing restrictions of the form \prod_{r\in R}\prod_{x\in S(b,r)}\rho(\sigma_{1}(\alpha,x))=1 on elements \alpha\in\Phi^{k}(F)\le\mathrm{Aut}(B_{d,k}).

<a id="X7F4CC068860BA22F" name="X7F4CC068860BA22F"></a>

<h5>4.4-1 SignHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignHomomorphism</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the sign homomorphism from <var class="Arg">F</var> to S_{2}.

The argument of this method is a permutation group <var class="Arg">F</var> \le S_{d}. This method can be used as an example for the argument <var class="Arg">rho</var> in the methods <code class="func">SpheresProduct</code> (<a href="chap4.html#X83A7A23D875BFAA2">4.4-3</a>) and <code class="func">LocalActionPi</code> (<a href="chap4.html#X79341499795BF8D9">4.4-4</a>).

<div class="example"><pre>
gap> F:=SymmetricGroup(3);;
gap> sign:=SignHomomorphism(F);
MappingByFunction( Sym( [ 1 .. 3 ] ), Sym( [ 1 .. 2 ] ), function( g ) ... end )
gap> Image(sign,(2,3));
(1,2)
gap> Image(sign,(1,2,3));
()
</pre></div>

<a id="X7D8D1FC07BC90971" name="X7D8D1FC07BC90971"></a>

<h5>4.4-2 AbelianizationHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianizationHomomorphism</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the homomorphism from <var class="Arg">F</var> to F/[F,F].

The argument of this method is a permutation group <var class="Arg">F</var> \le S_{d}. This method can be used as an example for the argument <var class="Arg">rho</var> in the methods <code class="func">SpheresProduct</code> (<a href="chap4.html#X83A7A23D875BFAA2">4.4-3</a>) and <code class="func">LocalActionPi</code> (<a href="chap4.html#X79341499795BF8D9">4.4-4</a>).

<div class="example"><pre>
gap> F:=PrimitiveGroup(5,3);
AGL(1, 5)
gap> ab:=AbelianizationHomomorphism(PrimitiveGroup(5,3));
[ (2,3,4,5), (1,2,3,5,4) ] -> [ f1, <identity> of ... ]
gap> Elements(Range(ab));
[ <identity> of ..., f1, f2, f1*f2 ]
gap> StructureDescription(Range(ab));
"C4"
</pre></div>

<a id="X83A7A23D875BFAA2" name="X83A7A23D875BFAA2"></a>

<h5>4.4-3 SpheresProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpheresProduct</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">rho</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the product \prod_{r\in R}\prod_{x\in S(b,r)}<var class="Arg">rho</var>(\sigma_{1}(<var class="Arg">aut</var>,x))\in\mathrm{im}(<var class="Arg">rho</var>).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a radius <var class="Arg">k</var> \in\mathbb{N}, an automorphism <var class="Arg">aut</var> of B_{d,k} all of whose 1-local actions are in the domain of the homomorphism <var class="Arg">rho</var> from a subgroup of S_d to an abelian group, and a sublist <var class="Arg">R</var> of <code class="code">[0..k-1]</code>. This method is used in the implementation of <code class="func">LocalActionPi</code> (<a href="chap4.html#X79341499795BF8D9">4.4-4</a>).

<div class="example"><pre>
gap> rho:=SignHomomorphism(SymmetricGroup(3));;
gap> SpheresProduct(3,2,LocalActionElement(2,3,(1,2)),rho,[0]);
(1,2)
gap> SpheresProduct(3,2,LocalActionElement(2,3,(1,2)),rho,[0,1]);
()
</pre></div>

<div class="example"><pre>
gap> F:=PrimitiveGroup(5,3);
AGL(1, 5)
gap> rho:=AbelianizationHomomorphism(F);;
gap> Elements(Range(rho));
[ <identity> of ..., f1, f2, f1*f2 ]
gap> StructureDescription(Range(rho));
"C4"
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> aut:=Random(mt,F);
(1,4,3,5)
gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[2]);
<identity> of ...
gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[1,2]);
f1
gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[0,1,2]);
f2
</pre></div>

<a id="X79341499795BF8D9" name="X79341499795BF8D9"></a>

<h5>4.4-4 LocalActionPi</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPi</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">rho</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the group \Pi^{l}(<var class="Arg">F</var>,<var class="Arg">rho</var>,<var class="Arg">R</var>)=\{\alpha\in\Phi^{l}(F)\mid \prod_{r\in R}\prod_{x\in S(b,r)}<var class="Arg">rho</var>(\sigma_{1}(\alpha,x))=1\}\le\mathrm{Aut}(B_{d,l}).

The arguments of this method are a degree <var class="Arg">l</var> \in\mathbb{N}_{\ge 2}, a radius <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a permutation group <var class="Arg">F</var> \le S_d, a homomorphism \rho from <var class="Arg">F</var> to an abelian group that is surjective on every point stabilizer in <var class="Arg">F</var>, and a non-empty, non-zero subset <var class="Arg">R</var> of <code class="code">[0..l-1]</code> that contains l-1.

<div class="example"><pre>
gap> F:=LocalAction(5,1,PrimitiveGroup(5,3));
AGL(1, 5)
gap> rho1:=AbelianizationHomomorphism(F);;
gap> rho2:=SignHomomorphism(F);;
gap> LocalActionPi(3,5,F,rho1,[0,1,2]);
<permutation group with 4 generators>
gap> Index(LocalActionPhi(3,F),last);
4
gap> LocalActionPi(3,5,F,rho2,[0,1,2]);
<permutation group with 5 generators>
gap> Index(LocalActionPhi(3,F),last);
2
</pre></div>

<a id="X87FE512E7DB7346C" name="X87FE512E7DB7346C"></a>

<h4>4.5 Semidirect products</h4>

When a subgroup F\le\mathrm{Aut}(B_{d,k}) satisfies (C) and admits an involutive compatibility cocycle z (which is automatic when k=1) one can characterise the kernels K\le\Phi_{k}(F)\cap\ker(\pi_{k}) that fit into a z-split exact sequence 1\to K\to\Sigma(F,K)\to F\to 1 for some subgroup \Sigma(F,K)\le\mathrm{Aut}(B_{d,k+1}) that satisfies (C). This characterisation is implemented in this section.

<a id="X7F425DFC8760388F" name="X7F425DFC8760388F"></a>

<h5>4.5-1 CompatibleKernels</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleKernels</code>( <var class="Arg">d</var>, <var class="Arg">F</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">‣ CompatibleKernels</code>( <var class="Arg">F</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the list of kernels K\le\prod_{\omega\in\Omega}F_{\omega}\cong\ker\pi\le\mathrm{Aut}(B_{d,2}) that are preserved by the action <var class="Arg">F</var> \curvearrowright\prod_{\omega\in\Omega}F_{\omega}, a\cdot(a_{\omega})_{\omega}:=(aa_{a^{-1}\omega}a^{-1})_{\omega}.

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, and a permutation group <var class="Arg">F</var> \le S_{d}. The kernels output by this method are compatible with <var class="Arg">F</var> with respect to the standard cocycle (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5.html#X80ADE0E379590053">5.3-1</a>)) and can be used in the method <code class="func">LocalActionSigma</code> (<a href="chap4.html#X83920A2C7CC46AC9">4.5-2</a>).

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">z</var></dt>
<dd>Returns: the list of kernels K\le\Phi_{k}(F)\cap\ker(\pi_{k})\le\mathrm{Aut}(B_{d,k+1}) that are normalized by \Gamma_{z}(<var class="Arg">F</var>) and such that for all k\in K and \omega\in\Omega there is k_{\omega}\in K with \mathrm{pr}_{\omega}k_{\omega}=z(\mathrm{pr}_{\omega}k,\omega)^{-1}.

The arguments of this method are a local action <var class="Arg">F</var> \le\mathrm{Aut}(B_{d,k}) that satisfies (C) and an involutive compatibility cocycle <var class="Arg">z</var> of <var class="Arg">F</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5.html#X80ADE0E379590053">5.3-1</a>)). It can be used in the method <code class="func">LocalActionSigma</code> (<a href="chap4.html#X83920A2C7CC46AC9">4.5-2</a>).

</dd>
</dl>

<div class="example"><pre>
gap> CompatibleKernels(3,SymmetricGroup(3));
[ Group(()), Group([ (1,2)(3,4)(5,6) ]), Group([ (3,4)(5,6), (1,2)(5,6) ]),
 Group([ (5,6), (3,4), (1,2) ]) ]
</pre></div>

<div class="example"><pre>
gap> P:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(P);;
gap> F:=LocalActionPi(2,3,P,rho,[1]);;
gap> z:=InvolutiveCompatibilityCocycle(F);;
gap> CompatibleKernels(F,z);
[ Group(()), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) ]),
 Group([ (1,2)(3,4)(5,6)(7,8), (5,6)(7,8)(9,10)(11,12) ]),
 Group([ (5,6)(7,8), (1,2)(3,4), (9,10)(11,12) ]) ]
</pre></div>

<a id="X83920A2C7CC46AC9" name="X83920A2C7CC46AC9"></a>

<h5>4.5-2 LocalActionSigma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionSigma</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">K</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">‣ LocalActionSigma</code>( <var class="Arg">F</var>, <var class="Arg">K</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">K</var></dt>
<dd>Returns: the semidirect product \Sigma(<var class="Arg">F</var>,<var class="Arg">K</var>)\le\mathrm{Aut}(B_{d,2}).

The arguments of this method are a degree <var class="Arg">d</var> \in\mathbb{N}_{\ge 3}, a subgroup <var class="Arg">F</var> of S_{d} and a compatible kernel <var class="Arg">K</var> for <var class="Arg">F</var> (see <code class="func">CompatibleKernels</code> (<a href="chap4.html#X7F425DFC8760388F">4.5-1</a>)).

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">K</var>, <var class="Arg">z</var></dt>
<dd>Returns: the semidirect product \Sigma_{z}(<var class="Arg">F</var>,<var class="Arg">K</var>)\le\mathrm{Aut}(B_{d,k+1}).

The arguments of this method are a local action <var class="Arg">F</var> of \mathrm{Aut}(B_{d,k}) that satisfies (C) and a kernel <var class="Arg">K</var> that is compatible for <var class="Arg">F</var> with respect to the involutive compatibility cocycle <var class="Arg">z</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5.html#X80ADE0E379590053">5.3-1</a>) and <code class="func">CompatibleKernels</code> (<a href="chap4.html#X7F425DFC8760388F">4.5-1</a>)) of <var class="Arg">F</var>.

</dd>
</dl>

<div class="example"><pre>
gap> S3:=SymmetricGroup(3);;
gap> kernels:=CompatibleKernels(3,S3);
[ Group(()), Group([ (1,2)(3,4)(5,6) ]), Group([ (3,4)(5,6), (1,2)(5,6) ]),
 Group([ (5,6), (3,4), (1,2) ]) ]
gap> for K in kernels do Print(Size(LocalActionSigma(3,S3,K)),"\n"); od;
6
12
24
48
</pre></div>

<div class="example"><pre>
gap> P:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(P);;
gap> F:=LocalActionPi(2,3,P,rho,[1]);;
gap> z:=InvolutiveCompatibilityCocycle(F);;
gap> kernels:=CompatibleKernels(F,z);
[ Group(()), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) ]),
 Group([ (1,2)(3,4)(5,6)(7,8), (5,6)(7,8)(9,10)(11,12) ]),
 Group([ (5,6)(7,8), (1,2)(3,4), (9,10)(11,12) ]) ]
gap> for K in kernels do Print(Size(LocalActionSigma(F,K,z)),"\n"); od;
24
48
96
192
</pre></div>

<a id="X7A8A46F4856570BE" name="X7A8A46F4856570BE"></a>

<h4>4.6 PGL₂ over the p-adic numbers</h4>

Here, we implement functions to provide the k-local actions of the groups \mathrm{PGL}(2,\mathbb{Q}_{p}) and \mathrm{PSL}(2,\mathbb{Q}_{p}) acting on T_{p+1}. This section is due to Tasman Fell.

<a id="X7E80A4A27E8D846C" name="X7E80A4A27E8D846C"></a>

<h5>4.6-1 LocalActionPGL2Qp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPGL2Qp</code>( <var class="Arg">p</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the subgroup of \mathrm{Aut}(B_{p+1,k}) induced by the action of \mathrm{PGL}(2,\mathbb{Z}_{p}) on the ball of radius <var class="Arg">k</var> around the vertex corresponding to the identity lattice of the Bruhat-Tits tree of \mathrm{PGL}(2,\mathbb{Q}_{p}).

The arguments of this method are a prime <var class="Arg">p</var> and a radius <var class="Arg">k</var> \in\mathbb{N}_{\ge 1}.

<div class="example"><pre>
gap> LocalActionPGL2Qp(3,1)=SymmetricGroup(4);
true
gap> F:=LocalActionPGL2Qp(5,3);; Size(F);
1875000
gap> SatisfiesC(F);
true
</pre></div>

<a id="X87D6A7B679C668A8" name="X87D6A7B679C668A8"></a>

<h5>4.6-2 LocalActionPSL2Qp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPSL2Qp</code>( <var class="Arg">p</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the subgroup of \mathrm{Aut}(B_{p+1,k}) induced by the action of \mathrm{PSL}(2,\mathbb{Z}_{p}) on the ball of radius <var class="Arg">k</var> around the vertex corresponding to the identity lattice of the Bruhat-Tits tree of \mathrm{PGL}(2,\mathbb{Q}_{p}).

The arguments of this method are a prime <var class="Arg">p</var> and a radius <var class="Arg">k</var> \in\mathbb{N}_{\ge 1}.

<div class="example"><pre>
gap> LocalActionPSL2Qp(3,1)=AlternatingGroup(4);
true
gap> F:=LocalActionPSL2Qp(5,3);; Size(F);
937500
gap> SatisfiesC(F);
true
</pre></div>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap3.html">[Previous Chapter]</a> <a href="chap5.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="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

¤ Dauer der Verarbeitung: 0.25 Sekunden (vorverarbeitet) ¤

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.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.