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<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>

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<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%">

‣ CompatibleKernels( d, F ) ( operation )

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