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< =>&t;</pan> span">:=estrictedClassFunction(k )^co1, g );;</span>

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<p><a id="X87A0A15D8588D62F" name="X87A0A15D8588D62F"></a></p>
<div class="ChapSects"><a href="chap8_mj.html#X87A0A15D8588D62F">8 <span class="Heading">Orbits, stabilisers and actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X81E0FF0587C54543">8.1 <span class="Heading">Orbits</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7CF168A0818C553D">8.1-1 FiningOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X822D1CFF7DDC11D7">8.1-3 FiningOrbitsDomain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7EAB52F67B3A0003">8.2 <span class="Heading">Stabilisers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X83B7F30F81A0BCE3">8.2-1 FiningStabiliser</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X854826AC7DA9FB6B">8.2-2 FiningStabiliserOrb</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7929900F79B6000C">8.2-3 FiningSetwiseStabiliser</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7A625B4D7992EEBC">8.2-4 StabiliserGroupOfSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7F50965D7B9C1BC5">8.2-5 ProjectiveStabiliserGroupOfSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X81B7F825830CD669">8.2-6 SpecialProjectiveStabiliserGroupOfSubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7B449F3B7F23A30A">8.3 <span class="Heading">Actions and nice monomorphisms revisited</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X86A646FF8668D82E">8.3-1 <span class="Heading">Action functions</span></a>
</pan
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X8474367181BB501E">8.3-2 <span class="Heading">Generic GAP functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7965086E82ABCF41">8.3-3 </re>/div>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7B47BE0983E93A83">8.3-4 NiceObject</a></span>
<span class="ContSS"><br
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X80F4EDD487059D70">8.3-6 SetParent</a></span>
</div></div>
</div>

<h3>8 <pan =java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 71

<p><a id="X81E0FF0587C54543" name="X81E0FF0587C54543"></a></p>

<h4>8.1 <span class="Heading">Orbits</span></h4>

<p>GAP provides generic functionality to compute orbits. These functions are, generally spoken, applicable to the groups implemented in <strong class="pkg">FinInG</strong>, combined with the appropriate action functions. However, the generic functions java.lang.StringIndexOutOfBoundsException: Index 256 out of bounds for length 80

<p><a id="X7CF168A0818C553D" name="X7CF168A0818C553D"></a></p>

<h5>8.1-1 FiningOrbit</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiningOrbit</code>( <var class="Arg">g</var>, <var class="Arg">obj</var>, <var class="Arg">act</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: Thespan class=GAPprompt"gap&t;</pan>span =GAPinput"factorders:=OrdersClassRepresentativesgmodz);<>

<p>The argument <var class="Arg">obj</var> is either a subspace of a projective space, then combined with the action function <code class="file">OnProjSubspaces</code>, or a set of elements of a projective space, then combined with the action function <code class="file">OnSetsProjSubspaces</code>span class"GAPprompt"gap><span>< class">hibar:=[;<span


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(6,3);</span>
Q(6, 3)
<span class="GAPprompt">span class="GAPprompt">></span> <span class="GAPinput">     if factorders[i] mod 2 = 1 then</span>
PGammaO(7,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(6,3);</span>
ProjectiveSpace(6, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">s := First(Solids(pg),t -> TypeOfSubspace(ps,t) = "elliptic" );</span>
<a  in ProjectiveSpace(6,,3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">orbit := FiningOrbit(g,s,OnProjSubspaces);</span>
<closed orbit, 265356 points>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
3555
 
</pre></div>

<p>The second example shows the possible use of <code class="file">FiningOrbit</code> in combination with the action function <code class="file">OnSetsProjSubspaces</code>. Please note that this variant is probably not the most efficient way to compute all elliptic quadrics contained in the parabolic quadric <span class="SimpleMath">\(ps\)</span>. Experiments show that for <span class


<div class="example"><pre>
<pan class=APprompt"gap&</java.lang.StringIndexOutOfBoundsException: Range [40, 37) out of bounds for length 97
Q(4, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(ps);</span>
span="GAPprompt">gt;/pan><pan class=GAPinput">od<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(4,3);</span>
ProjectiveSpace(4, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">s := First(Solids(pg),t -> torso:= phibar class=GAPprompt"gap><span><pan =APinput"cand:=(gmodz, rec(torso: phibar));/pan>
<a solid in ProjectiveSpace(4, 3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">orbit1 := FiningOrbit(g,s,OnProjSubspaces);</span>
<losedorbit, 36points>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
9
<span class="GAPprompt">gap></span> <span class="GAPinput">spts := Filtered(Points(s),s->s in ps);</span>
[ <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, 
  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, 
  <<pre>/iv>
  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, 
  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<closed orbit, 36 points>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
18
 
</pre></div>

<p><a id="X827CD5727A7DA4A3" name="X827CD5727A7DA4A3"></a></p>

<h5>8.1-2 FiningOrbits</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiningOrbits</code>( <var class="Arg">g</var>, <var class="Arg">set</var>, <var class="java.lang.StringIndexOutOfBoundsException: Range [0, 191) out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><divclass="">pre>
<p>Returns: The orbits of the group <var class="Arg">g</var> on <var class="Arg">set</var> under the action of <var class="Arg">act</var>.</p>

<p>The set is a set of elements of a projective space, the group <var class="Arg">g</var> is a subgroup of the collineation group of a projective space, and <var class="Arg">act</var> is the function <code class="file">OnProjSubspaces</code>. If <var class="Arg">coll</var> is a collection of elements of a projective space (i.e. notspan class="GAPprompt">gap></span> <span class="GAPinput">pi:= pi^m;;</span>


<div class="example"><pre>
<spanspan class=G">ap><span><java.lang.StringIndexOutOfBoundsException: Range [50, 44) out of bounds for length 112
H(3, 3^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(ps);</span>
PGammaU(4,<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 107
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningOrbits(g,Lines(PG(3,9)));</span>
75%..98%..100%..[ <closed orbit, 5670 points>, <closed orbit, 1680 points>, 
  an class="GAPprompt">></span> <span class="GAPinput">     if morders[i] mod 3 = 0 and phi[ PowerMap( m, 3 )[i] ] <> 0 then</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningOrbits(g,Planes(PG(3<pan =GAPprompt">gt;</pan><pan class=GAPinput"      Unbind( cand[] ;<spanjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
65%..100%..[ <closed orbit, 540 points>, <closed orbit, 280 points> ]
<put">ps :=(,)<span>
Q(2, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(ps);</span>
PGammaO(3,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">pts := Filtered(Points(PG(2,5)),x->not x in ps);;</span class="">< <pan class=GAPinput"   od;/pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(pts);</span>
25
<span class="GAPprompt">gap></span> <span class="pre>div>
48%..67%..100%..[ <closed orbit, 15 points>, <closed orbit, 6 points>, 
  <closed orbit, 10 points> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningOrbits(g,pts,OnProjSubspaces);</span>
60%..100%..[ <closed orbit, 15 points>, <closed orbit, 10 points> ]
 
</pre></div>

<p><a id="X822D1CFF7DDC11D7" name="X822D1CFF7DDC11D7"></a></p>

<h5>8.1-3 FiningOrbitsDomain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiningOrbitsDomain</code>( <var class="Arg">g</var>, <var class="Arg">coll</var>, <var class="Arg">act</var> )</td><td class="tdright">( operation 
<Returns: Theorbitsof the group<varclass=Arg"></>on  collection var =Arg">oll<var  theaction of<ar ="">ct/ar><p>

<p>The argument <var class="Arg">coll</var> must be an object in the category <code class="code">IsElementsOfIncidenceGeometry</code>, the argument <var class="Arg">g</var> can be any group, acting on the elements of <var class="Arg">coll</var> through a suitable action function <var class="Arg">act</var>. This operation is inspired by the GAP operation <code class="java.lang.StringIndexOutOfBoundsException: Index 373 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(7,2);</span>
ProjectiveSpace(7, 2)
<span class="GAPprompt">gap></span> <span class="GAPinput">group:=CollineationGroup(pg);</span>
The FinInG collineation group PGL(8,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">syl127:=SylowSubgroup(group,127);</span>
<projective collineation group of size 127>
<span class="GAPprompt">gap></span> <span class="GAPinput">orbits := FiningOrbits(syl127,AsList(Solids(pg)));;</span>
1%..2%..3%..4%..5%..6%..7%..8%..9%..10%..11%..12%..13%..14%..15%..16%..17%..18%..19%..20%..21%..
22%..23%..24%..25%..26%..27%..28%..29%..30%..31%..32%..33%..34%..35%..36%..37%..38%..39%..40%..41%
..2%..43%..44%..45%.46%..47%..%.49%..50%..51..52%.53%..54%..55%..56%..57%..58%..9..60%..
61%..62%..63%..64%..65%..66%..67%..68%..69%..70%..71%..72%..73%..74%..75%..76%..77%..78%..79%..
80%..81%..82%..83%<pan class="APprompt">gt;<span><pan class=GAPinput"                       &;ScalarProduct( m, chi, pi ) <> 0 );;</span>
..97%..98%..99%..100%..
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
212661
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected(List(orbits,x->Length(x)));</span>
[ [ 127, 1581 ] ]
<span class="GAPprompt">gap></</pre></div>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
26529
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected(List(orbits,x->Length(x)));</span>
[ [ 127, 1581 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
35506
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected(List(orbits,x->Length(x)));</span>
[ [ 127, 1581 ] ]
<span class=GAPprompt"gap&t;</span> <pan ="GAPinput">g :=AG(4,5);</span>
AG(4, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">h := Random(CollineationGroup(ag));</span>
< a collineation: <cmat 5x5 over GF(5,1)>, F^0>
<span class="GAPprompt"
<projective collineation group with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">orbits := FiningOrbitsDomain(group,Points(ag),OnAffineSubspaces);</span>
[ <closed orbit, 624 points>, <closed orbit, 1 points> ]
 
</pre></div>

<p><a id="X7EAB52F67B3A0003" name="X7EAB52F67B3A0003"></a></p>

<h4>8.2 <span class="Heading">Stabilisers</span></h4>

<p>The GAP function <code class="file">Stabilizer</code> is a generic function to compute stabilisers of one object (or sets or tuples etc. of objects) under a group, using a specified action function. This generic function can be used together with the in <strong class="pkg">FinInG</strong> implemented groups and elements of geometries. However, computing time can be very long, already in smallspan ="GAPprompt">ap&;<span> <panclass="GAPinput"cand:PermChars(m,<span


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(3,8);</span>
ProjectiveSpace(3, 8)
<span class="GAPprompt">gap&<spanclass="GAPprompt">></span> <span class="GAPinput">          lower:= ShallowCopy( pi - 1/3 * phi ),</span>
The FinInG collineation group PGammaL(4,8)
<span class="GAPprompt">gap></span> <span class<pan class=GAPprompt">gt;/pan><span =APinputnormalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ],</span>
<a point in ProjectiveSpace(3, 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,p,OnProjSubspaces);</span>
&t;rojective collineation group of size 177223237632  with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
10026
<span class="GAPprompt">gap></span> <span class="GAPinput">line := Random(Lines(ps));</span>
<a   (3 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,line,OnProjSubspaces);</span>
<projective collineation group of size 21849440256 with 2 generators>
<span class="GAPprompt">gap></span> <span class=" [ 16009115629875684006343550944921875,7774182899642733721875 
78126
 
</pre></div>

<p>The packages <strong class="pkg">GenSS</strong> and <strong class="pkg">orb</strong> java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 58

<p><a id="X83B7F30F81A0BCE3" name="X83B7F30F81A0BCE3"></a></p>

<h5>8.2-1 FiningStabiliser</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiningStabiliser</code>( <var class="Arg">g</var>, <var class="Arg">el</var> )</td><td class="tdright">( 1851609375,0,4680311220,405405,78624756 14467005 178605 
<p>Returns: The subgroup of <var class="Arg">g</var> stabilising the element <var class="Arg">el</var></p>

<p>The argument <var class="Arg">g</var> is a group of collineations acting on the element <var class="Arg">el</var>, being a subspace of a projective space (and hence, all elements of a Lie geometry are allowed as second argument). This operation relies on the <strong class="pkg">GenSS</strong> operation <code class="file">Stab</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(5,4);</span>
ProjectiveSpace(5, 4)
<pan class="GAPprompt">gap></span> <span class="GAPinput">g := SpecialHomographyGroup(ps);</span>
The FinInG PSL group PSL(6,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(ps));</span>
<a point in ProjectiveSpace(5, 4)>
<span class="GAPprompt">gap></21, 0, 0, 0, 0, 0, 98, 74, 42 0, ,0 120,76 10,0,0,0,java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
<projective collineation group of size 264696069567283200 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">line0 0, 1, 1, 0,6,0, 0,0,0,0 0 0, 0, 0, 0 0,0,5,3, 0,java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
< line inProjectiveSpace(5, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,line);</span>
<projective collineation group of size 3881174040576000 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">plane := Random(Planes(ps));</span>
<a plane in ProjectiveSpace(5, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,plane);</span>
#I  Have 106048 points.
#I  Have 158748 points.
<projective collineation group of size 958878292377600 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HyperbolicQuadric(5,5);</span>
Q+(5, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := IsometryGroup(ps);</span>
PGO(1,6,5)
<panclass="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(ps));</span>
<a point in Q+(5, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,p);</span>
<projective collineation group of size 36000000 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">line := Random(Lines(ps));</span>
<a line in Q+(5, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,line);</span>
<projective collineation group of size 6000000 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 31
<a plane in Q+(5, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,plane);</span>
<projective collineation group of size 93000000 with 2 generators>
<span class="GAPprompt">gap></span> <span class
H(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(h);</span>
#I  for Split Cayley Hexagon
#I  Computing nice monomorphism...
#I  Found permutation domain...
G_2(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(h));</span>
<a point in H(3)><>There is only one candidate, so we have found the permutation character.</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,p);</span>
<projective collineation group of size 11664 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">line :=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<a line in H(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliser(g,line);</span>
&collineation of size 11664 with 2generators>
 
</pre></div>

<p><a id="X854826AC7DA9FB6B" name="X854826AC7DA9FB6B"></a></p>

<h5>8.2-2 FiningStabiliserOrb</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiningStabiliserOrb</code>( <var class="Arg">g</var>, <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></
<p>Returns: The subgroup of <var class="Arg">g</var> stabilising the element <var class="Arg">el</var></p>

<p>The argument <var class="Arg">g</var> is a group of collineations acting on the element <var class="Arg">el</var>, being a subspace of a projective space (and hence, all elements of a Lie geometry are allowed as second argument). This operation relies on some particular <strongspan class=GAPprompt"gap&t;<span><panclass=GAPinput>: ("^211+2)(24S3)");<span>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(5,4);</span>
ProjectiveSpace(, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := SpecialHomographyGroup(ps);</span>
The FinInG PSL group PSL(6,4)
<span class="GAPprompt">gap></span> <span span class="GAPprompt">></span> <span class="GAPinput">                             [ TrivialCharacter( h ) ], map )[1] );</span>
<a point in ProjectiveSpace(5, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(g,p);</span>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">line := Random(Lines(ps));</span>
</pre></div>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(g,line);</span>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">plane := Random(Planes(ps));</span>
<a plane in ProjectiveSpace(5, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(g,plane);</span>
< collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HyperbolicQuadric(5,5);</span>
Q+(5, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := IsometryGroup(ps);</span>
PGO(1,6,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(ps));</span>
<a point in Q+(5, 5)>
<span class="GAPprompt">gap></span><h5>8.16- <span class="Heading">he Subgroup<span class=SimpleMath">(^3.2^6.2^{}2^{18}.(L_3(2)\times 3.S_6)\<span>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">line := Random(Lines(ps));</span>
<a line in Q+(5, 5)>
<span(September /span><h5>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">plane := Random(Planes(ps));</span>
<a plane in Q+(5, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(g,plane);</span>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := SplitCayleyHexagon(3);</span>
H(3)
<panclass="GAPprompt">gap></span><span class="GAPinput">g := CollineationGroup(h);</span>
#I  for Split Cayley Hexagon
#I  Computing nice monomorphism...
#I  Found permutation domain...
G_2(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(h));</span>
<a point in H(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(g,p);</span>
< group with 15 enerators>
<span class="GAPprompt">gap></span> <span class="GAPinput">line := Random(Lines(h));</span>
<a line in H(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(g,line);</span>
<projective collineation group with 15 generators>
 
</pre

<p>A small example shows the difference in computing time. Clearly the <code class="file">FiningStabiliserOrb</code> is the fastest way to compute stabilizers of one element.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(3,8);</span>
ProjectiveSpace(3, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(ps);</span>
The FinInG collineation group PGammaL(4,8)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(ps
<a point in ProjectiveSpace(3, 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := Stabilizer(g,p);</span>
&t;collineationgroup ofsize 177223237632 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
9576
<span class="GAPprompt">gap></span> <span class="GAPinput">g2 := FiningStabiliser(g,p);</span>
<projective collineation group of size 177223237632 with 2 span ="GAPprompt">gap></span> <span class="GAPinput">co1:= CharacterTable( "Co1" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
244
<span class=GAPprompt"gap></pan> <pan class=GAPinput">g3 :=FiningStabiliserOrb(g,p);</span>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
46
<span class="GAPprompt">gap></span> <span class="GAPinput">g1=g2;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g2=g3;</span>
true
 
</pre><

<p><a id="X7929900F79B6000C" name="X7929900F79B6000C"></a></p>

<h5>8.2-3 FiningSetwiseStabiliser</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiningSetwiseStabiliser</code>( <var class="Arg">g</var>, <var class="Arg">els</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The subgroup of <var class="Arg">g</var> stabilising the set <var class="Arg">els</var></p>

<p>The argument <var class=span class=GAPprompt"">gap&t;</pan><pan class=GAPinput">ist( filt,t-gt; Size(t)  order )<span>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HyperbolicQuadric(5,5);                   </span>
Q+(5, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := IsometryGroup(ps);</span>
PGO(1,6,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">plane1 := Random(Planes(ps));</span>
<a plane in Q+(5, 5)>
< class=GAPprompt">ap>;/span> spanclass="GAPinput">lane2 := Random(Planes(ps);</pan>
<a plane in Q+(5, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningSetwiseStabiliser(g,Set([plane1,plane2]));</span>
#  Computing adjusted stabilizer chain...
<projective collineation group with 4 generators>
 
</pre></div>

<p>Computing the setwise stabiliser under a group is also possible using <code class="file">Stabilizer</code>. But, not surprisingly, the[  


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span <span class="GAPprompt">gap></span> <span class="APprompt"gap&;span> <span class="GAPinput">k:= filt[2];;</span>
ProjectiveSpace(3, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(ps));</span>
<a point in ProjectiveSpace(3, 4)&</re>/iv>
<span class="GAPprompt">gap></span> <span class="GAPinput">q := Random(Points(ps));</span>
<a point in ProjectiveSpace(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(ps);</span>
The FinInG collineation group PGammaL(4,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(g,Set([p,q]),OnSets);</span>
<<>The list of maximal subgroups of <span class="SimpleMath">\(Co_1\)</span> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 183]</a>) tells us that <span class="SimpleMath">\(NU / N\)</span> is a maximal subgroup <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(Co_1\)</span> and has the structure <span class="SimpleMath">\(2^{{4+12}}.(S_3 \times 3.S_6)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pan =">>/> <span class="GAPinput">time;</span>
10440
 
</pre></div>

<p>The package <strong class="pkg">GenSS</strong> provides an efficient operations to compute setwise stabilisers. This is why <strong class="pkg">FinInG</strong> provides functionality, such as <code class="file">FiningSetwiseStabiliser</code>, to use these <strong class="pkg">GenSS</strong> operations for the particular groups and (elements) of geometries. A small example shows the difference in computing time.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(4,4);</span>
Q(4 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(ps);</span>
PGammaO(5,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">l1 := Random(Lines(ps));</span>
<a line in Q(4, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">l2 := Random(Lines(ps));</span>
<a line in Q(4, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput
<projective collineation group of size 2304 with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
2633
<span class="GAPprompt">gap></span> <span class="GAPinput">g2 := FiningSetwiseStabiliser(g,Set([l1,l2]));</span>
#I  Computing adjusted span class="GAPprompt">gap></>< class=GAPinput"f: CharacterTable( "(2)");<>
<projective collineation group with 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
70
<span class="GAPprompt">gap></span> <span class="GAPinput">g1=g2;</span>
true
 
</re</iv>

<p>For subspaces of projective spaces we can be even more efficient by writing down generators of the stabiliser and then return the generated subgroup.</p>

<p><a id="X7A625B4D7992EEBC" name="X7A625B4D7992EEBC"></a></p>

<h5>8.2-4 StabiliserGroupOfSubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; StabiliserGroupOfSubspace</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The subgroup of the collineation group of <var class="Arg">AmbientSpace(sub)</var> stabilising the subspace <var class="Arg">sub</var></p>

<p>The argument <var class="Arg">sub</var> is a subspace of a projective space.</p>

<p><a id="X7F50965D7B9C1BC5" name="X7F50965D7B9C1BC5"></a></p>

<h5>8.2-5 ProjectiveStabiliserGroupOfSubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveStabiliserGroupOfSubspace</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></trCharacter( CharacterTable( "L3(2)" ), [ 42, 6, 0, 0, 0, 0 ] ) ]
<p>Returns: The subgroup of the projectivity group of <var class="Arg">AmbientSpace(sub)</var> stabilising the subspace <var class="Arg">sub</var></p>

<p>The argument <var[Character( CharacterTable( "L3(2)" ), [ 168, 0, 0, 0, 0, 0 ] ) ]

<p><a id="X81B7F825830CD669" name="X81B7F825830CD669"></a></p>

<h5>8.2-6 SpecialProjectiveStabiliserGroupOfSubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SpecialProjectiveStabiliserGroupOfSubspace</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The subgroup of the special projectivity group of <var class="Arg">AmbientSpace(sub)</var> stabilising the subspace <var class="Arg">sub</var></p>

<p>The argument <var class="Arg">sub</var> is a subspace of a projective space.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg:=ProjectiveSpace(5,9);            </span>
ProjectiveSpace(5, 9)
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<a plane in ProjectiveSpace(5, 9)>
<span class="GAPprompt">gap></span> <span class="GAPinput">coll:=StabiliserGroupOfSubspace(sub); time;                                </span>
lt;projective collineation group of size 11173786189009966655078400 with 
6 generators>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">computed:=FiningStabiliserOrb(CollineationGroup(pg),sub); </span>
<projective collineation group with 15 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
34923
<span pThe character table of spanclass="impleMath"G\)/pan>is available in <trongclass="pkg">GAP</strong>, so we can compute the permutation character <span class="SimpleMath">\(\pi = 1_U^G\)</span> by computing the primitive permutation character <span class="SimpleMath">\(1_K^{{Co_1}}\)</span>, identifying it with <span class="SimpleMath">\(1_{{U/N}}^{{G/N}}\)</span>, and then inflating this character to <span class="SimpleMath">\(G\)</span>.</p>
true
<panclass="APprompt"&;/java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 117
<projective collineation group of size 5586893094504983327539200 with 
5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">FiningStabiliserOrb(ProjectivityGroup(pg),sub)=proj; </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">time;      </span>
116113
<span class=CharacterTable( "M" )
<projective collineation group of size 2793446547252491663769600 with 
5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">specproj = rTable "MC2B" );<span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
65564
 
</pre></div>

<p>haracterTable( "21+24.o1" )

<h4>8.3 <span class="Heading">Actions and nice monomorphisms revisited</span></h4>

<>GAPprovides generic functionsto computeaction homomorphismsand  their  forarbitrary groups.Thesefunctions areapplicable onthe projectivegroups implementedin strongclass=pkg"FinInG/trong.<java.lang.StringIndexOutOfBoundsException: Index 216 out of bounds for length 216

<p><a id="X86A646FF8668D82E" name="X86A646FF8668D82E"></a></p>

<h5>8.3-1 <span class="Heading">Action functions</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnProjSubspaces</code>( <var class="Arg">el</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnProjSubspacesExtended</code>( <var class="Arg">el</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnSetsProjSubspaces</code>( <var class="Arg">set</var>, <var class="Arg">g</var> 
<p>Returns: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Let <var class="Arg">el</var> be an element of any Lie geometry, and <var div class=example"<re>

<p><a id="X8474367181BB501E" name="X8474367181BB501E"></a></p>

<h5>8.3-2 <span class="Heading">Generic GAP functions</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><< class=GAPprompt"gap></span> <span class="GAPinput">nsgsizes:= List( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) );;</span>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Action</code>( <var class="Arg">g</var>, <var class="Arg">S</var>, <var class="Arg">act</var> )</td><td classspan class=GAPpromptgap></pan>span class="n=nsg[Position(nsgsizes (k   6) ];<span>
<p><var class="Arg">g</var> is a projective group, <var class="Arg">S</var> is a set or a collection of elements, <var class="Arg">act</var> is an action function. <code class="file">Action</code> simply returns <code class="file">Image(hom)</code>, if <var class="Arg">hom</var> is the result of <code class="file">ActionHomomorphism</code>. The examples are self-explanatory.</p>


<div class="example"><pre>
<class=GAPprompt"gap&t;/pan><pan class=GAPinput"pg = G(2,);/pan>
ProjectiveSpace(2, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">conic := Set(Points(ParabolicQuadric(2,3)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">coll := CollineationGroup(pg);</span>
The FinInG collineation group PGL(3,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">orb := Orbit(coll,conic,OnSetsProjSubspaces);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(orb);</span>
234
<span class="<span class="GAPpromptgap&;/> span java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 110
<action homomorphism>
<span class="GAPprompt">gap></span> <span class="GAPinput">perm := Image(hom);</span>
<permutation group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(perm);</span>
5616
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(perm);</span>
234
<pan class="GAPprompt">&;<span>< class=GAPinput"ps:=SymplecticSpace(,)</java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 96
W(5, 2)
<span class="GAPprompt">gap></span> <span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
PGammaSp(6,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">perm := Action(coll,Lines
<permutation group with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(perm);</span>
315

</pre></div>

<>Amonomorphism ofa group<span class=SimpleMath"(G\</pan> roughlyjust a java.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 300

<p><a id="X7965086E82ABCF41" name="X7965086E82ABCF41"></a></p>

<h5>8.3-3 NiceMonomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NiceMonomorphism</code>( <var class="Arg">group</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A group homomorphism</p>

<p>This is a generic GAP function, and returns a  to a" representation.</

<p><a id="X7B47BE0983E93A83" name="X7B47BE0983E93A83"></a></p>

<h5>8.3-4 NiceObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NiceObject</code>( <var class="Arg">group</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A permutation group</p>

<p><var class="Arg">group</var> is a projective group. The object this operation returns is actually equivalent with <code class="file">Image(NiceMonomorphism(group))</code>.</p>

<p><a id="X86AC831981D89DF1" name="X86AC831981D89DF1"></a></p>

<h5>8.3-5 <span class="Heading">Different behaviour for different collineation groups</span></h5>

<p>For the different Lie geometries implemented in <strong class="pkg">FinInG</strong>, nicemonomorphisms are (necessarily) treated in a different way. As the aim of a nicemonomorphism of group <span class="SimpleMath">\(G\)</span> is to provide a permutation representation

<p>For the collineation group, projectivity group and special projectivity group of a projective space, it is clear that the smallest degree permutation representation is the action of the group on the projective points. In principle, one could also consider the action on java.lang.StringIndexOutOfBoundsException: Index 275 out of bounds for length 119


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := CollineationGroup(PG(5,9));</span>
The FinInG collineation group PGammaL(6,9)
<class=GAPprompt">ap&t;span><spanclass=GAPinput">time;</pan>
28
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceMonomorphism(g);</span>
alse
<span class="GAPprompt">gap></span> <span class="GAPinput">h := CollineationGroup(EllipticQuadric(5,9));</span>
PGammaO-(6,9)
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
1584
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceMonomorphism(h);</span>
true

</pre></div>

<p><a id="X80F4EDD487059D70" name="X80F4EDD487059D70"></a></p>

<h5>8.3-6 SetParent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SetParent</code>( <var class="Arg">group</var> )</td><td class="tdright">( operation )</td></tr></table></div<span class="GAPprompt""&t;</span><span class=GAPinput">     elif psibar[i] = 0 then</span>
<p>Assume that <span class="SimpleMath">\(G\)</span> is a group of collineations. As mentioned already, from the underlying field and dimension, only the underlying projective space can be determined. An operation like <code class="file">Order</code> requires a nice monomorphism, so for an java.lang.StringIndexOutOfBoundsException: Index 293 out of bounds for length 74


<divspan class="APprompt">></span> <span class="GAPinput">     upperphibar[i]:= 2 * psibar[i];</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HermitianPolarSpace(3,81);</span>
H(, ^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">group := CollineationGroup(ps);</span>
PGammaU(4,9^2)
<span class=GAPprompt"gap><span> <pan class="GAPinput"time;<span>
2219
<span class="GAPprompt">gap></span> <span class="GAPinput">g := Random(group);</span>
< a collineation: <cmat 4x4 over GF(3,4)>, F^27>
<pan ="GAPprompt">gap></span> <span class="GAPinput">h := Random(group);</pan>
< a collineation: <cmat 4x4 over GF(3,4)>, F^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">group2 := Group([g,h]);</span>
<projective collineation group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceMonomorphism(group2);</span>
false
<panclass=GAPprompt"gap&t</><pan class=GAPinput"Order(group2)<span>
407194345728000
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
371559
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceMonomorphism(group2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(NiceObject(group2));</span>
538084
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Points(PG(3,81)));</span>
538084
<span class="GAPprompt">gap></span> <span class="GAPinput">group2 := Group([g,h]);</span>
<projective collineation group with 2 generators>
<span ="">ap></pan><panclass=GAPinput"SetParent(group2,group)</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceMonomorphism(group2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceObject(group2);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(group2);</span>
407194345728000
<n class=GAPprompt"gapgt;/pan> span class=">ime<span>
888
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNiceObject(group2);</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(NiceObject(group2));</span>
7300
< class=GAPprompt">gap&t;<span> span class=GAPinput">ize(Lines();<span
7300

</pre></div>


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