Quelle chap7.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/rcwa/doc/chap7.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 (RCWA) - Chapter 7: 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="chap7" onload="jscontent()">

<div class="chlinktop">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="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="chap6.html">[Previous Chapter]</a> <a href="chap8.html">[Next Chapter]</a> </div>

<a href="chap7_mj.html">[MathJax on]</a>
<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap7.html#X7A489A5D79DA9E5C">7 Examples</a>
<div class="ContSect"> <a href="chap7.html#X84A058CF7C65A908">7.1 
 Thompson's group V
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X86C2BAE3876985A6">7.2 
 Factoring Collatz' permutation of the integers
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X811919107D5DAAC1">7.3 
 The 3n+1 group
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X7DCFDC797FF213C5">7.4 
 A group with huge finite orbits
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X7968C1DF7EF0BD8E">7.5 
 A group which acts 4-transitively on the positive integers
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X85C529088050BEA3">7.6 
 A group which acts 3-transitively, but not 4-transitively on ℤ
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X878499AF7889FD9E">7.7 
 An rcwa mapping which seems to be contracting, but very slow
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X84A915BA833E0BDE">7.8 Checking a result by P. Andaloro</a>

</div>
<div class="ContSect"> <a href="chap7.html#X7E8CD9B67ED78735">7.9 Two examples by Matthews and Leigh</a>

</div>
<div class="ContSect"> <a href="chap7.html#X854E9F65817E4F63">7.10 Orders of commutators</a>

</div>
<div class="ContSect"> <a href="chap7.html#X7F085B867D799293">7.11 
 An infinite subgroup of CT(GF(2)[x]) with many torsion elements
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X7A8605E680F664BF">7.12 An abelian rcwa group over a polynomial ring</a>

</div>
<div class="ContSect"> <a href="chap7.html#X78DFE4B4821E07A6">7.13 Checking for solvability</a>

</div>
<div class="ContSect"> <a href="chap7.html#X783D54DC7A646273">7.14 Some examples over (semi)localizations of the integers</a>

</div>
<div class="ContSect"> <a href="chap7.html#X846D7D087861E0AC">7.15 
 Twisting 257-cycles into an rcwa mapping with modulus 32
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X78D5DC93845CA6A0">7.16 The behaviour of the moduli of powers </a>

</div>
<div class="ContSect"> <a href="chap7.html#X855A3CD88459958B">7.17 Images and preimages under the Collatz mapping </a>

</div>
<div class="ContSect"> <a href="chap7.html#X84B6A498838A5509">7.18 
 An extension of the Collatz mapping T to a permutation of ℤ^2
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X81EB8D397898C6B2">7.19 
 Finite quotients of Grigorchuk groups
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X7DD9502F80364631">7.20 
 Forward orbits of a monoid with 2 generators
</a>

</div>
<div class="ContSect"> <a href="chap7.html#X815800ED820C6ECF">7.21 
 The free group of rank 2 and the modular group PSL(2,ℤ)
</a>

</div>
</div>

<h3>7 Examples</h3>

This chapter discusses a number of examples of rcwa mappings and -groups in detail. All of them show different aspects of the package, and the order in which they appear is entirely arbitrary. In particular they are not ordered by degree of difficulty or interest.

The rcwa mappings, rcwa groups and other objects defined in this chapter can be found in the file <code class="file">pkg/rcwa/examples/examples.g</code>. This file can be read into the current GAP session by the function <code class="func">LoadRCWAExamples</code> (<a href="chap6.html#X8714254784AFD64B">6.1-1</a>) which takes no arguments and returns the name of a variable which the record containing the examples got assigned to. The global variable assignments made in a section of this chapter can be made by applying the function <code class="code">AssignGlobals</code> to the respective component of the examples record. The component names are given at the end of the corresponding sections.

The discussions of the examples are typically far from being exhaustive. It is quite likely that in many instances by just a few little modifications or additional easy commands you can find out interesting things yourself -- have fun!

<a id="X84A058CF7C65A908" name="X84A058CF7C65A908"></a>

<h4>7.1 
 Thompson's group V
</h4>

Thompson's group V, also known as Higman-Thompson group, is a finitely presented infinite simple group. This group has been found by Graham Higman, cf. [Hig74]. We show that the group

<div class="example"><pre>

gap> G := Group(List([[0,2,1,4],[0,4,1,4],[1,4,2,4],[2,4,3,4]],
> ClassTransposition));
<(0(2),1(4)),(0(4),1(4)),(1(4),2(4)),(2(4),3(4))>

</pre></div>

is isomorphic to Thompson's group V. This isomorphism has been pointed out by John P. McDermott. We take a slightly different set of generators:

<div class="example"><pre>

gap> k := ClassTransposition(0,2,1,2);;
gap> l := ClassTransposition(1,2,2,4);;
gap> m := ClassTransposition(0,2,1,4);;
gap> n := ClassTransposition(1,4,2,4);;
gap> H := Group(k,l,m,n);
<(0(2),1(2)),(1(2),2(4)),(0(2),1(4)),(1(4),2(4))>
gap> G = H; # k, l, m and n generate G as well
true

</pre></div>

Now we verify that our four generators satisfy the relations given on page 50 in <a href="chapBib.html#biBHigman74">[Hig74]</a>, when we read <code class="code">k</code> as κ, <code class="code">l</code> as λ, <code class="code">m</code> as μ and <code class="code">n</code> as ν:

<div class="example"><pre>

gap> HigmanThompsonRels :=
> [ k^2, l^2, m^2, n^2, # (1) in Higman's book
> l*k*m*k*l*n*k*n*m*k*l*k*m, # (2) "
> k*n*l*k*m*n*k*l*n*m*n*l*n*m, # (3) "
> (l*k*m*k*l*n)^3, (m*k*l*k*m*n)^3, # (4) "
> (l*n*m)^2*k*(m*n*l)^2*k, # (5) "
> (l*n*m*n)^5, # (6) "
> (l*k*n*k*l*n)^3*k*n*k*(m*k*n*k*m*n)^3*k*n*k*n,# (7) "
> ((l*k*m*n)^2*(m*k*l*n)^2)^3, # (8) "
> (l*n*l*k*m*k*m*n*l*n*m*k*m*k)^4, # (9) "
> (m*n*m*k*l*k*l*n*m*n*l*k*l*k)^4, #(10) "
> (l*m*k*l*k*m*l*k*n*k)^2, #(11) "
> (m*l*k*m*k*l*m*k*n*k)^2 ]; #(12) "
[ IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ) ]

</pre></div>

We conclude that our group is an homomorphic image of Thompson's group V. But since Thompson's group V is simple and our group is not trivial, this means indeed that the two groups are isomorphic.

In fact it is straightforward to show that <code class="code">G</code> is the group <code class="code">CT([2],Integers)</code> which is generated by the set of all class transpositions which interchange residue classes modulo powers of 2. First we check that <code class="code">G</code> contains all 11 class transpositions which interchange residue classes modulo 2 or 4:

<div class="example"><pre>

gap> S := Filtered(List(ClassPairs(4),ClassTransposition),
> ct->Mod(ct) in [2,4]);
[ ( 0(2), 1(2) ), ( 0(2), 1(4) ), ( 0(2), 3(4) ), ( 0(4), 1(4) ),
 ( 0(4), 2(4) ), ( 0(4), 3(4) ), ( 1(2), 0(4) ), ( 1(2), 2(4) ),
 ( 1(4), 2(4) ), ( 1(4), 3(4) ), ( 2(4), 3(4) ) ]
gap> IsSubset(G,S);
true

</pre></div>

Then we give a function which takes a class transposition τ ∈ CT_∅(ℤ), and which returns a factorization of an element γ satisfying τ^γ ∈ S into g_1 := τ_0(2),1(4) ∈ S, g_2 := τ_0(2),3(4) ∈ S, g_3 := τ_1(2),0(4) ∈ S, g_4 := τ_1(2),2(4) ∈ S, h_1 := τ_0(4),1(4) ∈ S and h_2 := τ_1(4),2(4) ∈ S:

<div class="example"><pre>

ReducingConjugator := function ( tau )

 local w, F, g1, g2, g3, g4, h1, h2, h, cls, cl, r;

 g1 := ClassTransposition(0,2,1,4); h1 := ClassTransposition(0,4,1,4);
 g2 := ClassTransposition(0,2,3,4); h2 := ClassTransposition(1,4,2,4);
 g3 := ClassTransposition(1,2,0,4);
 g4 := ClassTransposition(1,2,2,4);

 F := FreeGroup("g1","g2","g3","g4","h1","h2");

 w := One(F); if Mod(tau) <= 4 then return w; fi;

 # Before we can reduce the moduli of the interchanged residue classes,
 # we must make sure that both of them have at least modulus 4.
 cls := TransposedClasses(tau);
 if Mod(cls[1]) = 2 then
 if Residue(cls[1]) = 0 then
 if Residue(cls[2]) mod 4 = 1 then tau := tau^g2; w := w * F.2;
 else tau := tau^g1; w := w * F.1; fi;
 else
 if Residue(cls[2]) mod 4 = 0 then tau := tau^g4; w := w * F.4;
 else tau := tau^g3; w := w * F.3; fi;
 fi;
 fi;

 while Mod(tau) > 4 do # Now we can successively reduce the moduli.
 if not ForAny(AllResidueClassesModulo(2),
 cl -> IsEmpty(Intersection(cl,Support(tau))))
 then
 cls := TransposedClasses(tau);
 h := Filtered([h1,h2],
 hi->Length(Filtered(cls,cl->IsSubset(Support(hi),cl)))=1);
 h := h[1]; tau := tau^h;
 if h = h1 then w := w * F.5; else w := w * F.6; fi;
 fi;
 cl := TransposedClasses(tau)[2]; # class with larger modulus
 r := Residue(cl);
 if r mod 4 = 1 then tau := tau^g1; w := w * F.1;
 elif r mod 4 = 3 then tau := tau^g2; w := w * F.2;
 elif r mod 4 = 0 then tau := tau^g3; w := w * F.3;
 elif r mod 4 = 2 then tau := tau^g4; w := w * F.4; fi;
 od;

 return w;
end;

</pre></div>

After assigning <code class="code">g1</code>, <code class="code">g2</code>, <code class="code">g3</code>, <code class="code">g4</code>, <code class="code">h1</code> and <code class="code">h2</code> appropriately, we obtain for example:

<div class="example"><pre>

gap> ReducingConjugator(ClassTransposition(3,16,34,256));
h2*g1*h1*g1*h1*g1*h1*g1*h2*g2*h2*g4*h2*g4*h2*g3
gap> gamma := h2*g1*h1*g1*h1*g1*h1*g1*h2*g2*h2*g4*h2*g4*h2*g3;
<rcwa permutation of Z with modulus 256>
gap> ct := ClassTransposition(3,16,34,256)^gamma;;
gap> IsClassTransposition(ct);;
gap> ct;
ClassTransposition(1,4,2,4)

</pre></div>

Thompson's group V can also be embedded in a natural way into CT(GF(2)[x]):

<div class="example"><pre>

gap> x := Indeterminate(GF(2));; SetName(x,"x");
gap> R := PolynomialRing(GF(2),1);;
gap> k := ClassTransposition(0,x,1,x);;
gap> l := ClassTransposition(1,x,x,x^2);;
gap> m := ClassTransposition(0,x,1,x^2);;
gap> n := ClassTransposition(1,x^2,x,x^2);;
gap> G := Group(k,l,m,n);
<rcwa group over GF(2)[x] with 4 generators>

</pre></div>

The correctness of this representation can likewise be verified by simply checking the defining relations given above.

Enter <code class="code">AssignGlobals(LoadRCWAExamples().HigmanThompson);</code> in order to assign the global variables defined in this section.

<a id="X86C2BAE3876985A6" name="X86C2BAE3876985A6"></a>

<h4>7.2 
 Factoring Collatz' permutation of the integers
</h4>

In 1932, Lothar Collatz mentioned in his notebook the following permutation of the integers:

<div class="example"><pre>

gap> Collatz := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);;
gap> Display(Collatz);

Rcwa mapping of Z with modulus 3

 /
 | 2n/3 if n in 0(3)
n |-> < (4n-1)/3 if n in 1(3)
 | (4n+1)/3 if n in 2(3)
 \

gap> ShortCycles(Collatz,[-50..50],50); # There are some finite cycles:
[ [ 0 ], [ -1 ], [ 1 ], [ 2, 3 ], [ -2, -3 ], [ 4, 5, 7, 9, 6 ],
 [ -4, -5, -7, -9, -6 ],
 [ 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66 ],
 [ -44, -59, -79, -105, -70, -93, -62, -83, -111, -74, -99, -66 ] ]

</pre></div>

The cycle structure of Collatz' permutation has not been completely determined yet. In particular it is not known whether the cycle containing 8 is finite or infinite. Nevertheless, the factorization routine included in this package can determine a factorization of this permutation into class transpositions, i.e. involutions interchanging two disjoint residue classes:

<div class="example"><pre>

gap> Collatz in CT(Integers); # `Collatz' lies in the simple group CT(Z).
true
gap> Length(Factorization(Collatz));
212

</pre></div>

Setting the Info level of <code class="code">InfoRCWA</code> equal to 2 (simply issue <code class="code">RCWAInfo(2);</code>) causes the factorization routine to display detailed information on the progress of the factoring process. For reasons of saving space, this is not done in this manual.

We would like to get a factorization into fewer factors. Firstly, we try to factor the inverse -- just like the various options interpreted by the factorization routine, this has influence on decisions taken during the factoring process:

<div class="example"><pre>

gap> Length(Factorization(Collatz^-1));
129

</pre></div>

This is already a shorter product, but can still be improved. We remember the <code class="code">mKnot</code>'s, of which the permutation mKnot(3) looks very similar to Collatz' permutation. Therefore it is straightforward to try to factor both <code class="code">mKnot(3)</code> and <code class="code">Collatz/mKnot(3)</code>, and to look whether the sum of the numbers of factors is less than 129:

<div class="example"><pre>

gap> KnotFacts := Factorization(mKnot(3));;
gap> QuotFacts := Factorization(Collatz/mKnot(3));;
gap> List([KnotFacts,QuotFacts],Length);
[ 59, 9 ]
gap> CollatzFacts := Concatenation(QuotFacts,KnotFacts);
[ ( 0(6), 4(6) ), ( 0(6), 5(6) ), ( 0(6), 3(6) ), ( 0(6), 1(6) ),
 ( 0(6), 2(6) ), ( 2(3), 4(6) ), ( 0(3), 4(6) ), ( 2(3), 1(6) ),
 ( 0(3), 1(6) ), ( 0(36), 35(36) ), ( 0(36), 22(36) ),
 ( 0(36), 18(36) ), ( 0(36), 17(36) ), ( 0(36), 14(36) ),
 ( 0(36), 20(36) ), ( 0(36), 4(36) ), ( 2(36), 8(36) ),
 ( 2(36), 16(36) ), ( 2(36), 13(36) ), ( 2(36), 9(36) ),
 ( 2(36), 7(36) ), ( 2(36), 6(36) ), ( 2(36), 3(36) ),
 ( 2(36), 10(36) ), ( 2(36), 15(36) ), ( 2(36), 12(36) ),
 ( 2(36), 5(36) ), ( 21(36), 28(36) ), ( 21(36), 33(36) ),
 ( 21(36), 30(36) ), ( 21(36), 23(36) ), ( 21(36), 34(36) ),
 ( 21(36), 31(36) ), ( 21(36), 27(36) ), ( 21(36), 25(36) ),
 ( 21(36), 24(36) ), ( 26(36), 32(36) ), ( 26(36), 29(36) ),
 ( 10(18), 35(36) ), ( 5(18), 35(36) ), ( 10(18), 17(36) ),
 ( 5(18), 17(36) ), ( 8(12), 14(24) ), ( 6(9), 17(18) ),
 ( 3(9), 17(18) ), ( 0(9), 17(18) ), ( 6(9), 16(18) ), ( 3(9), 16(18) ),
 ( 0(9), 16(18) ), ( 6(9), 11(18) ), ( 3(9), 11(18) ), ( 0(9), 11(18) ),
 ( 6(9), 4(18) ), ( 3(9), 4(18) ), ( 0(9), 4(18) ), ( 0(6), 14(24) ),
 ( 0(6), 2(24) ), ( 8(12), 17(18) ), ( 7(12), 17(18) ),
 ( 8(12), 11(18) ), ( 7(12), 11(18) ), PrimeSwitch(3)^-1,
 ( 7(12), 17(18) ), ( 2(6), 17(18) ), ( 0(3), 17(18) ),
 PrimeSwitch(3)^-1, PrimeSwitch(3)^-1, PrimeSwitch(3)^-1 ]
gap> Product(CollatzFacts) = Collatz; # Check.
true

</pre></div>

The factors <code class="code">PrimeSwitch(3)</code> are products of 6 class transpositions (cf. <code class="func">PrimeSwitch</code> (<a href="chap2.html#X861C74E97AE5DA3B">2.5-3</a>)).

Enter <code class="code">AssignGlobals(LoadRCWAExamples().CollatzlikePerms);</code> in order to assign the global variables defined in this section.

<a id="X811919107D5DAAC1" name="X811919107D5DAAC1"></a>

<h4>7.3 
 The 3n+1 group
</h4>

The following group acts transitively on the set of positive integers for which the 3n+1 conjecture holds and which are not divisible by 6:

<div class="example"><pre>

gap> a := ClassTransposition(1,2,4,6);;
gap> b := ClassTransposition(1,3,2,6);;
gap> c := ClassTransposition(2,3,4,6);;
gap> G := Group(a,b,c);
<(1(2),4(6)),(1(3),2(6)),(2(3),4(6))>
gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions();
"3CTsGroups6"
gap> 3CTsGroups6.Id3CTsGroup(G,3CTsGroups6.grps); # 'catalogue number' of G
44132

</pre></div>

To see this, consider the action of G on the <q>3n+1 tree</q>. The vertices of this tree are the positive integers for which the 3n+1 conjecture holds, and for every vertex n there is an edge from n to T(n), where T denotes the Collatz mapping

<center> <img src = "collatz.png" width = "342" height = "63" alt = "T: Z -> Z, n |-> (n/2 if n even, (3n+1)/2 if n odd)"/> </center>

(cf. Chapter <a href="chap1.html#X83A8C2927FAE2C23">1</a>). It is easy to check that for every vertex n, either a, b or c maps n to T(n), and that the other two generators either fix n or map it to one of its preimages under T. So the 3n+1 conjecture is equivalent to the assertion that the group G acts transitively on N ∖ 0(6). First let's have a look at balls of small radius about 1 under the action of G -- these consist of those numbers whose trajectory under T reaches 1 quickly:

<div class="example"><pre>

gap> Ball(G,1,5,OnPoints);
[ 1, 2, 4, 5, 8, 10, 16, 32, 64 ]
gap> Ball(G,1,10,OnPoints);
[ 1, 2, 3, 4, 5, 8, 10, 13, 16, 20, 21, 26, 32, 40, 52, 53, 64, 80, 85,
 128, 160, 170, 256, 320, 340, 341, 512, 1024, 2048 ]
gap> Ball(G,1,15,OnPoints);
[ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 21, 22, 23, 26, 32, 34,
 35, 40, 44, 45, 46, 52, 53, 64, 68, 69, 70, 75, 80, 85, 104, 106, 113,
 128, 136, 140, 141, 151, 160, 170, 208, 212, 213, 226, 227, 256, 272,
 277, 280, 301, 302, 320, 340, 341, 416, 424, 452, 453, 454, 512, 640,
 680, 682, 832, 848, 853, 904, 908, 909, 1024, 1280, 1360, 1364, 1365,
 1664, 1696, 1706, 1808, 1813, 1816, 2048, 2560, 2720, 2728, 4096,
 5120, 5440, 5456, 5461, 8192, 10240, 10880, 10912, 10922, 16384,
 32768, 65536 ]
gap> Ball(G,1,15,OnPoints:Spheres);
[ [ 1 ], [ 2, 4 ], [ 8 ], [ 16 ], [ 5, 32 ], [ 10, 64 ],
 [ 3, 20, 21, 128 ], [ 40, 256 ], [ 13, 80, 85, 512 ],
 [ 26, 160, 170, 1024 ], [ 52, 53, 320, 340, 341, 2048 ],
 [ 17, 104, 106, 113, 640, 680, 682, 4096 ],
 [ 34, 35, 208, 212, 213, 226, 227, 1280, 1360, 1364, 1365, 8192 ],
 [ 11, 68, 69, 70, 75, 416, 424, 452, 453, 454, 2560, 2720, 2728, 16384
 ],
 [ 22, 23, 136, 140, 141, 151, 832, 848, 853, 904, 908, 909, 5120,
 5440, 5456, 5461, 32768 ],
 [ 7, 44, 45, 46, 272, 277, 280, 301, 302, 1664, 1696, 1706, 1808,
 1813, 1816, 10240, 10880, 10912, 10922, 65536 ] ]
gap> List(Ball(G,1,50,OnPoints:Spheres),Length);
[ 1, 2, 1, 1, 2, 2, 4, 2, 4, 4, 6, 8, 12, 14, 17, 20, 26, 32, 43, 52,
 66, 81, 104, 133, 170, 211, 271, 335, 424, 542, 686, 873, 1096, 1376,
 1730, 2205, 2794, 3522, 4429, 5611, 7100, 8978, 11343, 14296, 18058,
 22828, 28924, 36532, 46146, 58399, 73713 ]
gap> FloatQuotientsList(last);
[ 2., 0.5, 1., 2., 1., 2., 0.5, 2., 1., 1.5, 1.33333, 1.5, 1.16667,
 1.21429, 1.17647, 1.3, 1.23077, 1.34375, 1.2093, 1.26923, 1.22727,
 1.28395, 1.27885, 1.2782, 1.24118, 1.28436, 1.23616, 1.26567, 1.2783,
 1.26568, 1.27259, 1.25544, 1.25547, 1.25727, 1.27457, 1.26712,
 1.26056, 1.25752, 1.26688, 1.26537, 1.26451, 1.26342, 1.26034,
 1.26315, 1.26415, 1.26704, 1.26303, 1.26317, 1.26553, 1.26223 ]
gap> Difference(Filtered([1..100],n->n mod 6 <> 0),Ball(G,1,40,OnPoints));
[ 27, 31, 41, 47, 55, 62, 63, 71, 73, 82, 83, 91, 94, 95, 97 ]
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);;
gap> List(last2,n->Length(Trajectory(T,n,[1])));
[ 71, 68, 70, 67, 72, 69, 69, 66, 74, 71, 71, 60, 68, 68, 76 ]

</pre></div>

It is convenient to define an epimorphism from the free group of rank 3 to G:

<div class="example"><pre>

gap> F := FreeGroup("a","b","c");
<free group on the generators [ a, b, c ]>
gap> phi := EpimorphismByGenerators(F,G);
[ a, b, c ] -> [ ( 1(2), 4(6) ), ( 1(3), 2(6) ), ( 2(3), 4(6) ) ]

</pre></div>

We can compute balls about 1 in G:

<div class="example"><pre>

gap> B := Ball(G,One(G),7:Spheres);;
gap> List(B,Length);
[ 1, 3, 6, 12, 24, 48, 96, 192 ]
gap> List(B[3],Order);
[ 12, infinity, infinity, infinity, infinity, 12 ]
gap> List(B[3],g->PreImagesRepresentative(phi,g));
[ b*a, c*b, c*a, b*c, a*c, a*b ]
gap> g := a*b;; Order(g);;
gap> Display(g);

Rcwa permutation of Z with modulus 18, of order 12

( 1(6), 8(36), 4(18), 2(12) ) ( 3(6), 20(36), 10(18) )
( 5(6), 32(36), 16(18) )

</pre></div>

Spending some more time to compute <code class="code">B := Ball(G,One(G),12:Spheres);;</code>, one can check that (ab)^12 is the shortest word in the generators of G which does not represent the identity in the free product of 3 cyclic groups of order 2, but which represents the identity in G. However, the group G has elements of other finite orders as well -- for example:

<div class="example"><pre>

gap> g := (b*a)^3*b*c;; Order(g);;
gap> Display(g);

Rcwa permutation of Z with modulus 36, of order 105

( 8(9), 16(18), 64(72), 256(288), 85(96), 128(144), 32(36) )
( 7(12), 11(18), 22(36) ) ( 5(18), 10(36), 40(144), 13(48),
 20(72) ) ( 1(24), 2(36), 4(72) ) ( 14(36), 28(72), 112(288),
 37(96), 56(144) )

gap> Order(a*c*b*a*b*c*a*c);
60

</pre></div>

With some more efforts, one finds that e.g. (abc)^2c^b has order 616, that (abc)^2b has order 2310, that (ab)^2a^ca^bc has order 27720, and that a(c(ab)^2)^2 has order 65520. Of course G has many elements of infinite order as well. Some of them have infinite cycles, like e.g.

<div class="example"><pre>

gap> g := b*c;;
gap> Display(g);

Rcwa permutation of Z with modulus 12

 /
 | 4n if n in 1(3)
 | 2n if n in 5(6)
n |-> < n/2 if n in 2(12)
 | n/4 if n in 8(12)
 | n if n in 0(3)
 \

gap> Sinks(g);
[ 4(12) ]
gap> Trajectory(g,last[1],10);
[ 4(12), 16(48), 64(192), 256(768), 1024(3072), 4096(12288),
 16384(49152), 65536(196608), 262144(786432), 1048576(3145728) ]
gap> Trajectory(g,4,20);
[ 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304,
 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184,
 68719476736, 274877906944, 1099511627776 ]

</pre></div>

Others seem to have only finite cycles. Some of these appear to have <q>on average</q> comparatively <q>short</q> cycles, like e.g.

<div class="example"><pre>

gap> g := a*b*a*c*b*c;
<rcwa permutation of Z with modulus 144>
gap> cycs := ShortCycles(g,[0..10000],100,10^20);;
gap> Difference([0..10000],Union(cycs));
[ ]
gap> Collected(List(cycs,Length));
[ [ 1, 2222 ], [ 3, 1945 ], [ 4, 1111 ], [ 5, 93 ], [ 6, 926 ],
 [ 7, 31 ], [ 8, 864 ], [ 9, 10 ], [ 10, 289 ], [ 11, 4 ], [ 12, 95 ],
 [ 13, 1 ], [ 14, 31 ], [ 16, 12 ], [ 18, 4 ], [ 20, 1 ] ]

</pre></div>

If the cycle of g containing some n ∈ ℤ is finite and has a certain length l, then there is some m ∈ ℤ such that for every k ∈ ℤ the cycle of g containing n + km has length l as well. Thus, in other words, every finite cycle of g <q>belongs to</q> a cycle of residue classes. (This is a special property of g which is not shared by every rcwa permutation -- cf. e.g. Collatz' permutation from Section 7.2.) We can find some of these infinitely many residue class cycles:

<div class="example"><pre>

gap> cycsrc := ShortResidueClassCycles(g,Mod(g),20);
[ [ 0(6) ], [ 3(6), 160(288), 20(36) ],
 [ 7(18), 352(864), 44(108), 28(72) ],
 [ 11(18), 544(864), 2896(4608), 362(576), 68(108), 88(144) ],
 [ 13(18), 640(864), 80(108), 52(72) ], [ 10(36) ], [ 34(36) ],
 [ 1(54), 64(2592), 8(324), 4(216), 16(1152), 2(144) ],
 [ 5(54), 256(2592), 1360(13824), 170(1728), 32(324), 40(432),
 208(2304), 26(288) ],
 [ 17(54), 832(2592), 4432(13824), 23632(73728), 2954(9216), 554(1728),
 104(324), 136(432) ],
 [ 37(54), 1792(2592), 224(324), 148(216), 784(1152), 98(144) ],
 [ 41(54), 1984(2592), 10576(13824), 1322(1728), 248(324), 328(432),
 1744(2304), 218(288) ],
 [ 53(54), 2560(2592), 13648(13824), 72784(73728), 9098(9216),
 1706(1728), 320(324), 424(432) ], [ 38(72), 58(108), 304(576) ],
 [ 62(72), 94(108), 496(576) ] ]
gap> List(cycsrc,Length);
[ 1, 3, 4, 6, 4, 1, 1, 6, 8, 8, 6, 8, 8, 3, 3 ]
gap> Sum(List(Flat(cycsrc),cl->1/Mod(cl)));
97459/110592
gap> Float(last); # about 88% 'coverage'
0.881248
gap> cycsrc := ShortResidueClassCycles(g,3*Mod(g),20);
[ [ 0(6) ], [ 3(6), 160(288), 20(36) ],
 [ 7(18), 352(864), 44(108), 28(72) ],
 [ 11(18), 544(864), 2896(4608), 362(576), 68(108), 88(144) ],
 [ 13(18), 640(864), 80(108), 52(72) ], [ 10(36) ], [ 34(36) ],
 [ 1(54), 64(2592), 8(324), 4(216), 16(1152), 2(144) ],
 [ 5(54), 256(2592), 1360(13824), 170(1728), 32(324), 40(432),
 208(2304), 26(288) ],
 [ 17(54), 832(2592), 4432(13824), 23632(73728), 2954(9216), 554(1728),
 104(324), 136(432) ],
 [ 37(54), 1792(2592), 224(324), 148(216), 784(1152), 98(144) ],
 [ 41(54), 1984(2592), 10576(13824), 1322(1728), 248(324), 328(432),
 1744(2304), 218(288) ],
 [ 53(54), 2560(2592), 13648(13824), 72784(73728), 9098(9216),
 1706(1728), 320(324), 424(432) ], [ 38(72), 58(108), 304(576) ],
 [ 62(72), 94(108), 496(576) ],
 [ 23(162), 1120(7776), 5968(41472), 746(5184), 140(972), 184(1296),
 976(6912), 5200(36864), 650(4608), 122(864) ],
 [ 35(162), 1696(7776), 9040(41472), 48208(221184), 257104(1179648),
 32138(147456), 6026(27648), 1130(5184), 212(972), 280(1296) ],
 [ 73(162), 3520(7776), 440(972), 292(648), 1552(3456), 8272(18432),
 1034(2304), 194(432) ],
 [ 77(162), 3712(7776), 19792(41472), 2474(5184), 464(972), 616(1296),
 3280(6912), 17488(36864), 2186(4608), 410(864) ],
 [ 89(162), 4288(7776), 22864(41472), 121936(221184), 650320(1179648),
 81290(147456), 15242(27648), 2858(5184), 536(972), 712(1296) ],
 [ 127(162), 6112(7776), 764(972), 508(648), 2704(3456), 14416(18432),
 1802(2304), 338(432) ],
 [ 14(216), 22(324), 112(1728), 592(9216), 74(1152) ],
 [ 86(216), 130(324), 688(1728), 3664(9216), 458(1152) ] ]
gap> List(cycsrc,Length);
[ 1, 3, 4, 6, 4, 1, 1, 6, 8, 8, 6, 8, 8, 3, 3, 10, 10, 8, 10, 10, 8, 5,
 5 ]
gap> Sum(List(Flat(cycsrc),Density));
5097073/5308416
gap> Float(last); # already about 96% 'coverage'
0.960187

</pre></div>

There are also some elements of infinite order whose cycles seem to be all finite, but <q>on average</q> pretty <q>long</q> -- e.g.

<div class="example"><pre>

gap> g := (b*a*c)^2*a;;
gap> Display(g);

Rcwa permutation of Z with modulus 288

 /
 | (16n-1)/3 if n in 1(3)
 | (9n+5)/4 if n in 3(24) U 11(24)
 | (27n+19)/4 if n in 15(24) U 23(24)
 | (3n+1)/4 if n in 5(24)
 | (n-3)/6 if n in 21(24)
 | (27n+29)/8 if n in 9(48) U 41(48)
 | (9n+7)/8 if n in 17(48) U 33(48)
 | (2n-7)/9 if n in 8(36)
n |-> < (4n-11)/9 if n in 32(36)
 | (27n+38)/8 if n in 14(48)
 | (3n+2)/8 if n in 26(48)
 | (9n+10)/8 if n in 38(48)
 | (3n+4)/4 if n in 20(72)
 | n/4 if n in 56(72)
 | (9n+14)/16 if n in 2(96)
 | (27n+58)/16 if n in 50(96)
 | n if n in 0(6)
 \

gap> List([1..100],n->Length(Cycle(g,n)));
[ 6, 1, 6, 6, 6, 1, 194, 6, 216, 26, 26, 1, 26, 194, 65, 26, 26, 1, 216,
 26, 6, 216, 46, 1, 640, 26, 70, 194, 216, 1, 70, 26, 216, 216, 26, 1,
 194, 216, 73, 26, 110, 1, 194, 216, 194, 111, 39, 1, 194, 640, 640,
 194, 26, 1, 171, 194, 204, 640, 216, 1, 111, 70, 91, 26, 194, 1, 216,
 216, 26, 111, 65, 1, 50, 194, 26, 216, 640, 1, 502, 26, 111, 40, 110,
 1, 26, 194, 385, 640, 88, 1, 100, 111, 65, 110, 416, 1, 171, 194, 194,
 640 ]
gap> Length(Cycle(g,25));
640
gap> Maximum(Cycle(g,25));
323270249684063829
gap> Length(Cycle(g,25855));
4751
gap> Maximum(Cycle(g,25855));
507359605810239426786254778159924369135184044618585904603866210104085
gap> cycs := ShortCycles(g,[0..50000],10000,10^100);;
gap> S := [0..50000];;
gap> for cyc in cycs do S := Difference(S,cyc); od;
gap> S; # no cycle containing some n in [0..50000] has length > 10000 
[ ]

</pre></div>

Taking a look at the lengths of the trajectories of the Collatz mapping T starting at the points in a cycle, we can see how a cycle of g goes <q>up and down</q> in the 3n+1 tree:

<div class="example"><pre>

gap> List(Cycle(g,25),n->Length(Trajectory(T,n,[1])));
[ 17, 21, 25, 29, 33, 31, 35, 34, 32, 33, 37, 41, 45, 44, 42, 39, 43,
 41, 45, 44, 42, 43, 40, 38, 35, 39, 37, 41, 40, 44, 48, 46, 50, 49,
 47, 48, 45, 42, 46, 44, 48, 47, 45, 46, 50, 49, 47, 43, 41, 38, 39,
 36, 34, 30, 27, 31, 29, 33, 32, 30, 31, 35, 33, 37, 36, 40, 39, 43,
 41, 45, 44, 42, 43, 47, 51, 55, 53, 57, 56, 54, 55, 59, 58, 62, 66,
 64, 68, 67, 65, 66, 63, 60, 64, 62, 66, 65, 63, 64, 68, 67, 65, 61,
 59, 56, 52, 49, 53, 51, 55, 54, 52, 53, 57, 55, 59, 58, 56, 57, 54,
 50, 48, 45, 49, 47, 51, 50, 54, 52, 56, 55, 53, 54, 58, 62, 66, 70,
 74, 72, 76, 75, 79, 83, 87, 91, 90, 94, 93, 97, 95, 99, 98, 96, 97,
 94, 91, 88, 85, 89, 87, 91, 90, 88, 89, 86, 84, 81, 85, 83, 87, 86,
 90, 94, 98, 97, 101, 105, 109, 107, 111, 110, 108, 109, 113, 117, 115,
 119, 118, 122, 126, 125, 123, 120, 124, 122, 126, 125, 123, 124, 121,
 119, 116, 117, 114, 111, 115, 113, 117, 116, 114, 115, 119, 123, 122,
 120, 117, 121, 119, 123, 122, 120, 121, 118, 116, 112, 110, 106, 103,
 107, 105, 109, 108, 106, 107, 111, 109, 113, 112, 116, 114, 118, 117,
 115, 116, 113, 110, 111, 108, 104, 102, 99, 103, 101, 105, 104, 108,
 106, 110, 109, 107, 108, 112, 111, 109, 105, 102, 103, 100, 98, 95,
 92, 96, 94, 98, 97, 95, 96, 93, 91, 88, 92, 90, 94, 93, 97, 101, 105,
 109, 108, 106, 103, 107, 105, 109, 108, 106, 107, 104, 102, 99, 103,
 101, 105, 104, 108, 112, 110, 114, 113, 111, 112, 116, 115, 113, 109,
 106, 110, 108, 112, 111, 109, 110, 114, 112, 116, 115, 113, 114, 111,
 107, 105, 102, 103, 100, 98, 95, 99, 97, 101, 100, 104, 103, 107, 105,
 109, 108, 106, 107, 104, 101, 98, 99, 96, 94, 91, 92, 89, 87, 84, 85,
 82, 80, 77, 81, 79, 83, 82, 86, 85, 89, 88, 86, 83, 80, 81, 78, 76,
 73, 74, 71, 68, 72, 70, 74, 73, 71, 72, 76, 80, 79, 83, 87, 91, 90,
 88, 85, 89, 87, 91, 90, 88, 89, 86, 84, 81, 85, 83, 87, 86, 90, 94,
 92, 96, 95, 93, 94, 98, 96, 100, 99, 97, 98, 102, 106, 110, 114, 113,
 111, 108, 112, 110, 114, 113, 111, 112, 109, 107, 104, 108, 106, 110,
 109, 113, 117, 115, 119, 118, 116, 117, 114, 111, 115, 113, 117, 116,
 114, 115, 119, 118, 116, 112, 110, 107, 108, 105, 103, 100, 104, 102,
 106, 105, 109, 108, 112, 110, 114, 113, 111, 112, 116, 115, 113, 109,
 106, 103, 104, 101, 99, 95, 91, 88, 92, 90, 94, 93, 91, 92, 96, 94,
 98, 97, 95, 96, 100, 98, 102, 101, 105, 104, 102, 99, 100, 97, 93, 89,
 87, 84, 85, 82, 80, 77, 74, 78, 76, 80, 79, 77, 78, 75, 73, 69, 67,
 64, 68, 66, 70, 69, 73, 71, 75, 74, 72, 73, 70, 67, 68, 65, 63, 60,
 64, 62, 66, 65, 69, 68, 66, 63, 64, 61, 59, 56, 60, 58, 62, 61, 65,
 64, 62, 59, 60, 57, 55, 51, 48, 49, 46, 44, 40, 37, 34, 35, 32, 28,
 26, 23, 27, 25, 29, 28, 32, 30, 34, 33, 31, 32, 36, 35, 33, 29, 26,
 27, 24, 22, 19, 23, 21, 25, 24, 28, 27, 25, 22, 23, 20, 18, 14, 18,
 22, 20, 24, 23, 21, 22, 19, 16, 20, 18, 22, 21, 19, 20, 24, 23, 21,
 17, 15, 17, 15, 19, 18, 16 ]
gap> lngs := List(Cycle(g,25855),n->Length(Trajectory(T,n,[1])));;
gap> Minimum(lngs);
55
gap> Maximum(lngs);
521
gap> Position(lngs,55);
15
gap> Position(lngs,521);
2807

</pre></div>

Finally let's have a look at elements of G with small modulus:

<div class="example"><pre>

gap> B := RestrictedBall(G,One(G),20,36:Spheres);;
gap> List(B,Length);
[ 1, 3, 6, 12, 4, 6, 6, 4, 4, 4, 6, 6, 3, 3, 2, 0, 0, 0, 0, 0, 0 ]
gap> Sum(last);
70
gap> Position(last2,0)-2;
14

</pre></div>

So we have 70 elements of modulus 36 or less in G which can be reached from the identity by successive multiplication with generators without passing elements with mudulus exceeding 36. Further we see that the longest word in the generators yielding an element with modulus at most 36 has length 14. Now we double our bound on the modulus:

<div class="example"><pre>

gap> B := RestrictedBall(G,One(G),100,72:Spheres);;
gap> List(B,Length);
[ 1, 3, 6, 12, 22, 14, 18, 22, 24, 26, 26, 34, 35, 32, 37, 38, 46, 58,
 65, 73, 82, 91, 93, 96, 110, 121, 114, 117, 146, 138, 148, 168, 174,
 196, 215, 214, 232, 255, 280, 305, 315, 359, 377, 371, 363, 366, 397,
 419, 401, 405, 405, 401, 407, 415, 435, 424, 401, 359, 338, 330, 332,
 281, 278, 271, 269, 254, 255, 257, 258, 258, 233, 215, 202, 185, 154,
 121, 88, 55, 35, 20, 10, 5, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0 ]
gap> Sum(last);
15614
gap> Position(last2,0)-2;
83
gap> Collected(List(Flat(B),Modulus));
[ [ 1, 1 ], [ 6, 3 ], [ 12, 4 ], [ 18, 2 ], [ 24, 4 ], [ 36, 56 ],
 [ 48, 4 ], [ 72, 15540 ] ]

</pre></div>

We observe that there are 15540 elements in G with modulus 72 which are <q>reachable</q> from the identity by successive multiplication with generators without passing elements with mudulus exceeding 72. Further we see that the longest word in the generators yielding an element with modulus at most 72 has length 83.

It is obvious that many questions regarding the group G remain open.

<a id="X7DCFDC797FF213C5" name="X7DCFDC797FF213C5"></a>

<h4>7.4 
 A group with huge finite orbits
</h4>

In this section we investigate a group which has huge finite orbits on ℤ.

<div class="example"><pre>

gap> a := ClassTransposition(0,2,1,2);;
gap> b := ClassTransposition(0,5,4,5);;
gap> c := ClassTransposition(1,4,0,6);;
gap> G := Group(a,b,c);
<(0(2),1(2)),(0(5),4(5)),(1(4),0(6))>
gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions();
"3CTsGroups6"
gap> 3CTsGroups6.Id3CTsGroup(G,3CTsGroups6.grps); # 'catalogue number' of G
1284

</pre></div>

We look for orbits of length at most 100 containing an integer in the range <code class="code">[0..1000]</code>:

<div class="example"><pre>

gap> orbs := ShortOrbits(G,[0..1000],100);;
gap> List(orbs,Length);
[ 16, 2, 24, 2, 2, 2, 8, 2, 8, 2, 2, 8, 2, 8, 2, 2, 2, 40, 2, 8, 24, 2,
 8, 2, 2, 8, 2, 24, 8, 2, 56, 2, 2, 2, 8, 2, 8, 2, 2, 8, 2, 8, 2, 2, 2,
 24, 2, 8, 2, 8, 2, 2, 8, 2, 8, 2, 24, 2, 2, 2, 8, 2, 8, 2, 2, 8, 2, 8,
 2, 2, 2, 2, 8, 24, 2, 8, 2, 2, 8, 2, 24, 8, 2, 2, 2, 2, 8, 2, 8, 2, 2,
 8, 2, 8, 2, 2, 2, 24, 2, 8, 2, 8, 2, 2, 8, 2, 8, 2, 24, 2, 2 ]
gap> Collected(last);
[ [ 2, 67 ], [ 8, 32 ], [ 16, 1 ], [ 24, 9 ], [ 40, 1 ], [ 56, 1 ] ]
gap> Length(Difference([0..1000],Union(orbs)));
491

</pre></div>

So almost half of the integers in the range <code class="code">[0..1000]</code> lie in orbits of length larger than 100. In fact there are much larger orbits. For example:

<div class="example"><pre>

gap> B := Ball(G,32,500,OnPoints:Spheres);; # compute ball about 32
gap> Position(B,[]); # <> fail -> we have exhausted the orbit
354
gap> Sum(List(B,Length)); # the orbit length
6296
gap> Maximum(Flat(B)); # the largest integer in the orbit
3301636381609509797437679
gap> B := Ball(G,736,5000,OnPoints:Spheres);; # the same for 736 ...
gap> Position(B,[]);
2997
gap> Sum(List(B,Length)); # the orbit length for this time
495448
gap> Maximum(Flat(B));
2461374276522713949036151811903149785690151467356354652860276957152301465\
0546360696627187194849439881973442451686685024708652634593861146709752378\
847078493406287854573381920553713155967741550498839

</pre></div>

It seems that the cycles of abc completely traverse all orbits of G, with the only exception of the orbit of 0. Let's check this in the above examples:

<div class="example"><pre>

gap> g := a*b*c;;
gap> Display(g);

Rcwa permutation of Z with modulus 60

 /
 | n-1 if n in 3(30) U 9(30) U 17(30) U 23(30) U 27(30) U
 | 29(30)
 | 3n/2 if n in 0(20) U 12(20) U 16(20)
 | n+1 if n in 2(20) U 6(20) U 10(20)
 | (2n+1)/3 if n in 7(30) U 13(30) U 19(30)
 | n+3 if n in 1(30) U 11(30)
n |-> < n-5 if n in 15(30) U 25(30)
 | (3n+12)/2 if n in 4(20)
 | (3n-12)/2 if n in 8(20)
 | n+5 if n in 14(20)
 | n-3 if n in 18(20)
 | (2n-7)/3 if n in 5(30)
 | (2n+9)/3 if n in 21(30)
 \

gap> Length(Cycle(g,32));
6296
gap> Length(Cycle(g,736));
495448

</pre></div>

Representatives and lengths of the cycles of g which intersect nontrivially with the range <code class="code">[0..1000]</code> are as follows:

<div class="example"><pre>

gap> CycleRepresentativesAndLengths(g,[0..1000]:notify:=50000);
n = 736: after 50000 steps, the iterate has 157 binary digits.
n = 736: after 100000 steps, the iterate has 135 binary digits.
n = 736: after 150000 steps, the iterate has 131 binary digits.
n = 736: after 200000 steps, the iterate has 507 binary digits.
n = 736: after 250000 steps, the iterate has 414 binary digits.
n = 736: after 300000 steps, the iterate has 457 binary digits.
n = 736: after 350000 steps, the iterate has 465 binary digits.
n = 736: after 400000 steps, the iterate has 325 binary digits.
n = 736: after 450000 steps, the iterate has 534 binary digits.
n = 896: after 50000 steps, the iterate has 359 binary digits.
n = 896: after 100000 steps, the iterate has 206 binary digits.
[ [ 1, 15 ], [ 2, 2 ], [ 16, 24 ], [ 22, 2 ], [ 26, 2 ], [ 32, 6296 ],
 [ 46, 2 ], [ 52, 8 ], [ 56, 296 ], [ 62, 2 ], [ 76, 8 ], [ 82, 2 ],
 [ 86, 2 ], [ 92, 8 ], [ 106, 2 ], [ 112, 104 ], [ 116, 8 ],
 [ 122, 2 ], [ 136, 440 ], [ 142, 2 ], [ 146, 2 ], [ 152, 40 ],
 [ 166, 2 ], [ 172, 8 ], [ 176, 24 ], [ 182, 2 ], [ 196, 8 ],
 [ 202, 2 ], [ 206, 2 ], [ 212, 8 ], [ 226, 2 ], [ 232, 24 ],
 [ 236, 8 ], [ 242, 2 ], [ 256, 56 ], [ 262, 2 ], [ 266, 2 ],
 [ 272, 408 ], [ 286, 2 ], [ 292, 8 ], [ 296, 104 ], [ 302, 2 ],
 [ 316, 8 ], [ 322, 2 ], [ 326, 2 ], [ 332, 8 ], [ 346, 2 ],
 [ 352, 264 ], [ 356, 8 ], [ 362, 2 ], [ 376, 1304 ], [ 382, 2 ],
 [ 386, 2 ], [ 392, 24 ], [ 406, 2 ], [ 412, 8 ], [ 416, 200 ],
 [ 422, 2 ], [ 436, 8 ], [ 442, 2 ], [ 446, 2 ], [ 452, 8 ],
 [ 466, 2 ], [ 472, 104 ], [ 476, 8 ], [ 482, 2 ], [ 496, 24 ],
 [ 502, 2 ], [ 506, 2 ], [ 512, 696 ], [ 526, 2 ], [ 532, 8 ],
 [ 536, 3912 ], [ 542, 2 ], [ 556, 8 ], [ 562, 2 ], [ 566, 2 ],
 [ 572, 8 ], [ 586, 2 ], [ 592, 888 ], [ 596, 8 ], [ 602, 2 ],
 [ 616, 728 ], [ 622, 2 ], [ 626, 2 ], [ 632, 2776 ], [ 646, 2 ],
 [ 652, 8 ], [ 656, 24 ], [ 662, 2 ], [ 676, 8 ], [ 682, 2 ],
 [ 686, 2 ], [ 692, 8 ], [ 706, 2 ], [ 712, 24 ], [ 716, 8 ],
 [ 722, 2 ], [ 736, 495448 ], [ 742, 2 ], [ 746, 2 ], [ 752, 1272 ],
 [ 766, 2 ], [ 772, 8 ], [ 776, 376 ], [ 782, 2 ], [ 796, 8 ],
 [ 802, 2 ], [ 806, 2 ], [ 812, 8 ], [ 826, 2 ], [ 832, 120 ],
 [ 836, 8 ], [ 842, 2 ], [ 856, 2264 ], [ 862, 2 ], [ 866, 2 ],
 [ 872, 24 ], [ 886, 2 ], [ 892, 8 ], [ 896, 132760 ], [ 902, 2 ],
 [ 916, 8 ], [ 922, 2 ], [ 926, 2 ], [ 932, 8 ], [ 946, 2 ],
 [ 952, 456 ], [ 956, 8 ], [ 962, 2 ], [ 976, 24 ], [ 982, 2 ],
 [ 986, 2 ], [ 992, 1064 ] ]

</pre></div>

So far the author has checked that all positive integers less than 173176 lie in finite cycles of g. Several of them are longer than 1000000, and the cycle containing 25952 has length 245719352. Whether the cycle containing 173176 is finite or infinite has not been checked so far -- in any case it is longer than 5700000000, and it exceeds 10^40000. Presumably it is finite as well, but checking this may require a lot of computing time.

On the one hand the cycles of g seem to behave <q>randomly</q>, perhaps as if they would ascend or descend from one point to the next by a certain factor depending on which side a thrown coin falls on. -- In this <q>model</q>, cycles would be finite with probability 1 since the simple random walk on ℤ is recurrent. On the other, there seems to be quite some structure on them, however little is known so far.

First we observe that each orbit under the action of G seems to split into two cycles of h := abcacb of the same length (of course this has been checked for many more orbits than those shown here):

<div class="example"><pre>

gap> h := a*b*c*a*c*b;
<rcwa permutation of Z with modulus 360>
gap> List(CyclesOnFiniteOrbit(G,h,32),Length);
[ 3148, 3148 ]
gap> List(CyclesOnFiniteOrbit(G,h,736),Length);
[ 247724, 247724 ]

</pre></div>

One cycle seems to contain the points at the odd positions and the other seems to contain the points at the even positions in the cycle of g:

<div class="example"><pre>

gap> cycle_g := Cycle(g,32);;
gap> positions1 := List(Cycle(h,32),n->Position(cycle_g,n));;
gap> Collected(positions1 mod 2);
[ [ 1, 3148 ] ]
gap> positions2 := List(Cycle(h,33),n->Position(cycle_g,n));;
gap> Collected(positions2 mod 2);
[ [ 0, 3148 ] ]

</pre></div>

However the ordering in which these points are traversed looks pretty <q>scrambled</q>:

<div class="example"><pre>

gap> positions1{[1..200]};
[ 1, 6271, 6291, 6281, 6285, 6287, 6283, 6289, 6273, 6275, 6277, 6279,
 6293, 5, 15, 17, 19, 6259, 6261, 6263, 6265, 21, 23, 25, 41, 6227,
 6229, 6231, 6233, 6235, 6237, 6239, 43, 53, 55, 57, 63, 59, 61, 65,
 45, 47, 49, 51, 67, 6223, 6221, 69, 6163, 6215, 6205, 6209, 6211,
 6207, 6213, 6165, 6171, 6177, 6179, 6181, 6183, 6175, 6173, 6185,
 6189, 6191, 6187, 6193, 6169, 6167, 6195, 6199, 6201, 6197, 6203,
 6217, 73, 83, 85, 87, 103, 113, 115, 117, 4357, 4361, 4363, 4359,
 4365, 4371, 4373, 4375, 4377, 4369, 4367, 4379, 119, 121, 123, 125,
 129, 131, 127, 133, 139, 141, 143, 145, 137, 135, 147, 149, 151, 153,
 155, 159, 161, 157, 163, 169, 175, 4283, 4281, 177, 4271, 4273, 4275,
 4277, 181, 4255, 4257, 4259, 4261, 4263, 4265, 4267, 183, 2161, 2163,
 4195, 4199, 4201, 4197, 4203, 4209, 4211, 4213, 4215, 4207, 4205,
 4217, 2165, 2167, 2169, 2171, 2175, 2177, 2173, 2179, 2185, 2187,
 2189, 2191, 2183, 2181, 2193, 2195, 2197, 2199, 2201, 2467, 2469,
 4117, 4121, 4123, 4119, 4125, 4131, 4133, 4135, 4137, 4129, 4127,
 4139, 2471, 2473, 2475, 2477, 2487, 2489, 2491, 2507, 2517, 2519,
 2521, 2537, 3923, 3925, 3941, 3943 ]

</pre></div>

<a id="X7968C1DF7EF0BD8E" name="X7968C1DF7EF0BD8E"></a>

<h4>7.5 
 A group which acts 4-transitively on the positive integers
</h4>

In this section, we would like to show that the group G generated by the two permutations

<div class="example"><pre>

gap> a := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]);;
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> SetName(a,"a"); SetName(u,"u"); G := Group(a,u);;

</pre></div>

which we have already investigated in earlier examples acts 4-transitively on the set of positive integers. Obviously, it acts on the set of positive integers. First we show that this action is transitive. We start by checking in which residue classes sufficiently large positive integers are mapped to smaller ones by a suitable group element:

<div class="example"><pre>

gap> List([a,a^-1,u,u^-1],DecreasingOn);
[ 1(2), 0(3), 0(5) U 2(5), 2(3) ]
gap> Union(last);
Z \ 4(30) U 16(30) U 28(30)

</pre></div>

We see that we cannot always choose such a group element from the set of generators and their inverses -- otherwise the union would be <code class="code">Integers</code>.

<div class="example"><pre>

gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2],DecreasingOn);
[ 1(2), 0(3), 0(5) U 2(5), 2(3), 1(8) U 7(8), 0(3) U 2(9) U 7(9),
 0(25) U 12(25) U 17(25) U 20(25), 2(3) U 1(9) U 3(9) ]
gap> Union(last); # Still not enough ...
Z \ 4(90) U 58(90) U 76(90)
gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a*u,u*a,(a*u)^-1,(u*a)^-1],
> DecreasingOn);
[ 1(2), 0(3), 0(5) U 2(5), 2(3), 1(8) U 7(8), 0(3) U 2(9) U 7(9),
 0(25) U 12(25) U 17(25) U 20(25), 2(3) U 1(9) U 3(9),
 3(5) U 0(10) U 7(20) U 9(20), 0(5) U 2(5), 2(3), 3(9) U 4(9) U 8(9) ]
gap> Union(last); # ... but that's it!
Integers

</pre></div>

Finally, we have to deal with <q>small</q> integers. We use the notation for the coefficients of rcwa mappings introduced at the beginning of this manual. Let c_r(m) > a_r(m). Then we easily see that (a_r(m)n+b_r(m))/c_r(m) > n implies n < b_r(m)/(c_r(m)-a_r(m)). Thus we can restrict our considerations to integers n < b_ max, where b_ max is the largest second entry of a coefficient triple of one of the group elements in our list:

<div class="example"><pre>

gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a*u,u*a,(a*u)^-1,(u*a)^-1],
> f->Maximum(List(Coefficients(f),c->c[2])));
[ 1, 1, 4, 2, 7, 7, 56, 28, 25, 17, 17, 11 ]
gap> Maximum(last);
56

</pre></div>

Thus this upper bound is 56. The rest is easy -- all we have to do is to check that the orbit containing 1 contains also all other positive integers less than or equal to 56:

<div class="example"><pre>

gap> S := [1];;
gap> while not IsSubset(S,[1..56]) do
> S := Union(S,S^a,S^u,S^(a^-1),S^(u^-1));
> od;
gap> IsSubset(S,[1..56]);
true

</pre></div>

Checking 2-transitivity is computationally harder, and in the sequel we will omit some steps which are in practice needed to find out <q>what to do</q>. The approach taken here is to show that the stabilizer of 1 in G acts transitively on the set of positive integers greater than 1. We do this by similar means as used above for showing the transitivity of the action of G on the positive integers. We start by determining all products of at most 5 generators and their inverses, which stabilize 1 (taking at most 4-generator products would not suffice!):

<div class="example"><pre>

gap> gens := [a,u,a^-1,u^-1];;
gap> tups := Concatenation(List([1..5],k->Tuples([1..4],k)));;
gap> Length(tups);
1364
gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]],
> l->PositionSublist(tup,l)=fail));;
gap> Length(tups);
484
gap> stab := [];;
gap> for tup in tups do
> n := 1;
> for i in tup do n := n^gens[i]; od;
> if n = 1 then Add(stab,tup); fi;
> od;
gap> Length(stab);
118
gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i])));;
gap> ForAll(stabelm,elm->1^elm=1); # Check.
true

</pre></div>

The resulting products have various different not quite small moduli:

<div class="example"><pre>

gap> List(stabelm,Modulus);
[ 4, 3, 16, 25, 9, 81, 64, 100, 108, 100, 25, 75, 27, 243, 324, 243,
 256, 400, 144, 400, 100, 432, 324, 400, 80, 400, 625, 25, 75, 135,
 150, 75, 225, 81, 729, 486, 729, 144, 144, 81, 729, 1296, 729, 6561,
 1024, 1600, 192, 1600, 400, 576, 432, 1600, 320, 1600, 2500, 100, 100,
 180, 192, 192, 108, 972, 1728, 972, 8748, 1600, 400, 320, 80, 1600,
 2500, 300, 2500, 625, 625, 75, 675, 75, 75, 135, 405, 600, 120, 600,
 1875, 75, 225, 405, 225, 225, 675, 243, 2187, 729, 2187, 216, 216,
 243, 2187, 1944, 2187, 19683, 576, 144, 576, 432, 81, 81, 729, 2187,
 5184, 324, 8748, 243, 2187, 19683, 26244, 19683 ]
gap> Lcm(last);
12597120000
gap> Collected(Factors(last));
[ [ 2, 10 ], [ 3, 9 ], [ 5, 4 ] ]

</pre></div>

Similar as before, we determine for any of the above mappings the residue classes whose elements larger than the largest b_r(m) - coefficient of the respective mapping are mapped to smaller integers:

<div class="example"><pre>

gap> decs := List(stabelm,DecreasingOn);;
--> --------------------

--> maximum size reached

--> --------------------

quality100%

¤ Dauer der Verarbeitung: 0.43 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 ist noch experimentell.