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<Chapter Label="ch:Examples"><Heading>Examples</Heading>

<Ignore Remark="set screen width to 75, for the example tester">
<Example>
<![CDATA[
gap> SizeScreen([75,24]);;
]]>
</Example>
</Ignore>

This chapter discusses a number of <Q>nice</Q> 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 interestingness or difficulty. <P/>

Please note that now since quite a while no further examples have been
added to this chapter, and that the capabilities of this package have been
extended considerably in the meantime. <P/>

The rcwa mappings defined in this chapter (and in fact many more) can be
found in the file <F>pkg/rcwa/examples/examples.g</F>, so there is no need
to extract them from the manual files. This file can be read into the current
&GAP; session by issueing <C>RCWAReadExamples( );</C>. <P/>

The examples are typically far from discussing the respective aspects
exhaustively. 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!

<!-- #################################################################### -->

<Section Label="sec:FactoringTheCollatzPermutation">
<Heading>
  Factoring Collatz' permutation of the integers
</Heading>

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

<Example>
<![CDATA[
gap> Collatz := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);;
gap> SetName(Collatz,"Collatz"); Display(Collatz);

Rcwa mapping of Z with modulus 3

              n mod 3               |             n^Collatz
------------------------------------+------------------------------------
  0                                 | 2n/3
  1                                 | (4n - 1)/3
  2                                 | (4n + 1)/3

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

The cycle structure of Collatz' permutation has not been completely
determined yet. In particular it is not known whether the cycle
containing&nbsp;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:

<Example>
<![CDATA[
gap> Collatz in CT(Integers);  # `Collatz' lies in the simple group CT(Z).
true
gap> Length(Factorization(Collatz));
212
]]>
</Example>

Setting the Info level of <C>InfoRCWA</C> equal to&nbsp;2 (simply issue
<C>RCWAInfo(2);</C>) 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. <P/>

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:

<Example>
<![CDATA[
gap> Length(Factorization(Collatz^-1));
129
]]>
</Example>

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

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

The factors <C>PrimeSwitch(3)</C> are products of 6 class transpositions
(cf.&nbsp;<Ref Func="PrimeSwitch" Label="p"/>).
At the end of Section <Ref Label="sec:WildButFiniteCycles"/>, a much
smaller factorization task is performed <Q>manually</Q> for purposes
of illustration.

</Section>

<!-- #################################################################### -->

<Section Label="sec:SlowlyContractingMapping">
<Heading>
  An rcwa mapping which seems to be contracting, but very slow
</Heading>

The iterates of an integer under the Collatz mapping <M>T</M> seem to
approach its contraction centre -- this is the finite set where all
trajectories end up after a finite number of steps -- rather quickly and
do not get very large before doing so (of course this is a purely heuristic
statement as the <M>3n+1</M> Conjecture has not been proved so far!):

<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);;
gap> S0 := LikelyContractionCentre(T,100,1000);
#I  Warning: `LikelyContractionCentre' is highly probabilistic.
The returned result can only be regarded as a rough guess.
See ?LikelyContractionCentre for more information.
[ -136, -91, -82, -68, -61, -55, -41, -37, -34, -25, -17, -10, -7, -5, 
  -1, 0, 1, 2 ]
gap> S0^T = S0; # This holds by definition of the contraction centre.
true
gap> List([1..30],n->Length(Trajectory(T,n,S0)));
[ 1, 1, 5, 2, 4, 6, 11, 3, 13, 5, 10, 7, 7, 12, 12, 4, 9, 14, 14, 6, 6, 
  11, 11, 8, 16, 8, 70, 13, 13, 13 ]
gap> Maximum(List([1..1000],n->Length(Trajectory(T,n,S0))));
113
gap> Maximum(List([1..1000],n->Maximum(Trajectory(T,n,S0))));
125252
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

The following mapping seems to be contracting as well, but its trajectories
are much longer:

<Log>
<![CDATA[
gap> f6 := RcwaMapping([[ 1,0,6],[ 5, 1,6],[ 7,-2,6],
>                       [11,3,6],[11,-2,6],[11,-1,6]]);;
gap> SetName(f6,"f6");
gap> Display(f6);

Rcwa mapping of Z with modulus 6

              n mod 6               |                n^f6
------------------------------------+------------------------------------
  0                                 | n/6
  1                                 | (5n + 1)/6
  2                                 | (7n - 2)/6
  3                                 | (11n + 3)/6
  4                                 | (11n - 2)/6
  5                                 | (11n - 1)/6

gap> S0 := LikelyContractionCentre(f6,1000,100000);;
#I  Warning: `LikelyContractionCentre' is highly probabilistic.
The returned result can only be regarded as a rough guess.
See ?LikelyContractionCentre for more information.
gap> Trajectory(f6,25,S0);
[ 25, 21, 39, 72, 12, 2 ]
gap> List([1..100],n->Length(Trajectory(f6,n,S0)));
[ 1, 1, 3, 4, 1, 2, 3, 2, 1, 5, 7, 2, 8, 17, 3, 16, 1, 4, 17, 6, 5, 2, 
  5, 5, 6, 1, 4, 2, 15, 1, 1, 3, 2, 5, 13, 3, 2, 3, 4, 1, 8, 4, 4, 2, 7, 
  19, 23517, 3, 9, 3, 1, 18, 14, 2, 20, 23512, 14, 2, 6, 6, 1, 4, 19, 
  12, 23511, 8, 23513, 10, 1, 13, 13, 3, 1, 23517, 7, 20, 7, 9, 9, 6, 
  12, 8, 6, 18, 14, 23516, 31, 12, 23545, 4, 21, 19, 5, 1, 17, 17, 13, 
  19, 6, 23515 ]
gap> Maximum(Trajectory(f6,47,S0));
7363391777762473304431877054771075818733690108051469808715809256737742295\
45698886054
]]>
</Log>

Computing the trajectory of 3224 takes quite a while -- this trajectory
ascends to about <M>3 \cdot 10^{2197}</M>, before it approaches the fixed
point&nbsp;2 after 19949562 steps. <P/>

When constructing the mapping <C>f6</C>, the denominators of the
partial mappings have been chosen to be equal and the numerators have
been chosen to be numbers coprime to the common denominator, whose product
is just a little bit smaller than the <C>Modulus(f6)</C>th power of the
denominator. In the example we have <M>5 \cdot 7 \cdot 11^3 = 46585</M>
and <M>6^6 = 46656</M>. <P/>

Although the trajectories of <C>T</C> are much shorter than those of
<C>f6</C>, it seems likely that this does not make the problem of deciding
whether the mapping&nbsp;<C>T</C> is contracting essentially easier --
even for mappings with much shorter trajectories than&nbsp;<C>T</C>
the problem seems to be equally hard. A solution can usually only be found
in trivial cases, i.e. for example when there is some <M>k</M> such that
applying the <M>k</M>th power of the respective mapping to any integer
decreases its absolute value.

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:AndaloroResult">
<Heading>Checking a result by P. Andaloro</Heading>

In <Cite Key="Andaloro00"/>, P.&nbsp;Andaloro has shown that proving that
trajectories of integers <M>n \in 1(16)</M> under the Collatz mapping always
contain&nbsp;1 would be sufficient to prove the <M>3n+1</M> Conjecture.
In the sequel, this result is verified by &RCWA;. Checking that the
union of the images of the residue class 1(16) under powers of the Collatz
mapping&nbsp;<M>T</M> contains <M>\mathbb{Z} \setminus 0(3)</M> is obviously
enough. Thus we put <M>S := 1(16)</M>, and successively unite the
set&nbsp;<M>S</M> with its image under&nbsp;<M>T</M>:

<Example>
<![CDATA[
gap> S := ResidueClass(Integers,16,1);
1(16)
gap> S := Union(S,S^T);
1(16) U 2(24)
gap> S := Union(S,S^T);
1(12) U 2(24) U 17(48) U 33(48)
gap> S := Union(S,S^T);
<union of 30 residue classes (mod 144)>
gap> S := Union(S,S^T);
<union of 42 residue classes (mod 144)>
gap> S := Union(S,S^T);
<union of 172 residue classes (mod 432)>
gap> S := Union(S,S^T);
<union of 676 residue classes (mod 1296)>
gap> S := Union(S,S^T);
<union of 810 residue classes (mod 1296)>
gap> S := Union(S,S^T);
<union of 2638 residue classes (mod 3888)>
gap> S := Union(S,S^T);
<union of 33 residue classes (mod 48)>
gap> S := Union(S,S^T);
<union of 33 residue classes (mod 48)>
gap> Union(S,ResidueClass(Integers,3,0)); # Et voila ...
Integers
]]>
</Example>

Further similar computations are shown in
Section&nbsp;<Ref Label="sec:CollatzImagesAndPreImages"/>.

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:MatthewsLeighExamples">
<Heading>Two examples by Matthews and Leigh</Heading>

In <Cite Key="MatthewsLeigh87"/>, K.&nbsp;R.&nbsp;Matthews and
G.&nbsp;M.&nbsp;Leigh have shown that two trajectories of the following
(surjective, but not injective) mappings are acyclic (mod&nbsp;<M>x</M>)
and divergent:

<Example>
<![CDATA[
gap> x := Indeterminate(GF(4),1);; SetName(x,"x");
gap> R := PolynomialRing(GF(2),1);
GF(2)[x]
gap> ML1 := RcwaMapping(R,x,[[1,0,x],[(x+1)^3,1,x]]*One(R));;
gap> ML2 := RcwaMapping(R,x,[[1,0,x],[(x+1)^2,1,x]]*One(R));;
gap> SetName(ML1,"ML1"); SetName(ML2,"ML2");
gap> Display(ML1);

Rcwa mapping of GF(2)[x] with modulus x

        P mod x         |                     P^ML1
------------------------+------------------------------------------------
 0*Z(2)                 | P/x
 Z(2)^0                 | ((x^3+x^2+x+Z(2)^0)*P + Z(2)^0)/x

gap> Display(ML2);

Rcwa mapping of GF(2)[x] with modulus x

        P mod x         |                     P^ML2
------------------------+------------------------------------------------
 0*Z(2)                 | P/x
 Z(2)^0                 | ((x^2+Z(2)^0)*P + Z(2)^0)/x

gap> List([ML1,ML2],IsSurjective);
[ true, true ]
gap> List([ML1,ML2],IsInjective);
[ false, false ]
gap> traj1 := Trajectory(ML1,One(R),16);
[ Z(2)^0, x^2+x+Z(2)^0, x^4+x^2+x, x^3+x+Z(2)^0, x^5+x^4+x^2, x^4+x^3+x, 
  x^3+x^2+Z(2)^0, x^5+x^2+Z(2)^0, x^7+x^6+x^5+x^3+Z(2)^0, 
  x^9+x^7+x^6+x^5+x^3+x+Z(2)^0, x^11+x^10+x^8+x^7+x^6+x^5+x^2, 
  x^10+x^9+x^7+x^6+x^5+x^4+x, x^9+x^8+x^6+x^5+x^4+x^3+Z(2)^0, 
  x^11+x^8+x^7+x^6+x^4+x+Z(2)^0, x^13+x^12+x^11+x^8+x^7+x^6+x^4, 
  x^12+x^11+x^10+x^7+x^6+x^5+x^3 ]
gap> traj2 := Trajectory(ML2,(x^3+x+1)*One(R),16);
[ x^3+x+Z(2)^0, x^4+x+Z(2)^0, x^5+x^3+x^2+x+Z(2)^0, x^6+x^3+Z(2)^0, 
  x^7+x^5+x^4+x^2+x, x^6+x^4+x^3+x+Z(2)^0, x^7+x^4+x^3+x+Z(2)^0, 
  x^8+x^6+x^5+x^4+x^3+x+Z(2)^0, x^9+x^6+x^3+x+Z(2)^0, 
  x^10+x^8+x^7+x^5+x^4+x+Z(2)^0, x^11+x^8+x^7+x^5+x^4+x^3+x^2+x+Z(2)^0, 
  x^12+x^10+x^9+x^8+x^7+x^5+Z(2)^0, x^13+x^10+x^7+x^4+x, 
  x^12+x^9+x^6+x^3+Z(2)^0, x^13+x^11+x^10+x^8+x^7+x^5+x^4+x^2+x, 
  x^12+x^10+x^9+x^7+x^6+x^4+x^3+x+Z(2)^0 ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

The pattern which Matthews and Leigh used to show the divergence of the
above trajectories can be recognized easily by looking at the corresponding
Markov chains with the two states 0&nbsp;mod&nbsp;<M>x</M> and
1&nbsp;mod&nbsp;<M>x</M>:

<Example>
<![CDATA[
gap> traj1modx := Trajectory(ML1,One(R),400,x);;
gap> traj2modx := Trajectory(ML2,(x^3+x+1)*One(R),600,x);;
gap> List(traj1modx{[1..150]},val->Position([Zero(R),One(R)],val)-1);
[ 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 
  1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 
  1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
gap> List(traj2modx{[1..150]},val->Position([Zero(R),One(R)],val)-1);
[ 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 
  1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ]
]]>
</Example>

What is important here are the lengths of the intervals between two changes
from one state to the other:

<Example>
<![CDATA[
gap> ChangePoints := l->Filtered([1..Length(l)-1],pos->l[pos]<>l[pos+1]);;
gap> Diffs := l->List([1..Length(l)-1],pos->l[pos+1]-l[pos]);;
gap> Diffs(ChangePoints(traj1modx)); # The pattern in the first ...
[ 1, 1, 2, 4, 2, 2, 4, 8, 4, 4, 8, 16, 8, 8, 16, 32, 16, 16, 32, 64, 32, 
  32, 64 ]
gap> Diffs(ChangePoints(traj2modx)); # ... and in the second example.
[ 1, 7, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 193, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
gap> Diffs(ChangePoints(last)); # Make this a bit more obvious.
[ 1, 3, 1, 7, 1, 15, 1, 31, 1, 63, 1 ]
]]>
</Example>

This looks clearly acyclic, thus the trajectories diverge.
Needless to say however that this computational evidence does not replace
the proof along these lines given in the article cited above, but just
sheds a light on the idea behind it.

</Section>

<!-- #################################################################### -->

<Section Label="sec:StructureOfWildGroup">
<Heading>Exploring the structure of a wild rcwa group</Heading>

In this example, a simple attempt to should be made to investigate the
structure of a given wild group by finding orders of torsion elements.
In general, determining the structure of a given wild group seems to be
a very hard task. First of all, the group in question has to be defined:

<Example>
<![CDATA[
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> SetName(u,"u");
gap> Display(u);

Rcwa mapping of Z with modulus 5

              n mod 5               |                n^u
------------------------------------+------------------------------------
  0                                 | 3n/5
  1                                 | (9n + 1)/5
  2                                 | (3n - 1)/5
  3                                 | (9n - 2)/5
  4                                 | (9n + 4)/5

gap> nu := ClassShift(0,1);;
gap> G := Group(u,nu);
<rcwa group over Z with 2 generators>
gap> IsTame(G);
false
]]>
</Example>

Now we would like to know which orders torsion elements of&nbsp;<C>G</C>
can have -- taking a look at the above generators it seems to make sense
to try commutators:

<Example>
<![CDATA[
gap> l := Filtered([0..100],k->IsTame(Comm(u,nu^k)));
[ 0, 2, 3, 5, 6, 9, 10, 12, 13, 15, 17, 18, 20, 21, 24, 25, 27, 28, 30, 
  32, 33, 35, 36, 39, 40, 42, 43, 45, 47, 48, 50, 51, 54, 55, 57, 58, 
  60, 62, 63, 65, 66, 69, 70, 72, 73, 75, 77, 78, 80, 81, 84, 85, 87, 
  88, 90, 92, 93, 95, 96, 99, 100 ]
gap> List(l,k->Order(Comm(u,nu^k)));
[ 1, 6, 5, 3, 5, 5, 3, infinity, 7, infinity, 7, 5, 3, infinity, 
  infinity, 3, 5, 7, infinity, 7, infinity, 3, 5, 5, 3, 5, infinity, 
  infinity, infinity, 5, 3, 5, 5, 3, infinity, 7, infinity, 7, 5, 3, 
  infinity, infinity, 3, 5, 7, infinity, 7, infinity, 3, 5, 5, 3, 5, 
  infinity, infinity, infinity, 5, 3, 5, 5, 3 ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> Display(Comm(u,nu^13));

Bijective rcwa mapping of Z with modulus 9

              n mod 9               |                n^f
------------------------------------+------------------------------------
  0 3 6                             | n + 5
  1 4 7                             | 3n - 9
  2 8                               | n - 11
  5                                 | (n + 16)/3

gap> Order(Comm(u,nu^13));
7
gap> u2 := u^2;
<wild bijective rcwa mapping of Z with modulus 25>
gap> Filtered([1..16],k->IsTame(Comm(u2,nu^k))); # k < 15 -> commutator wild!
[ 15 ]
gap> Order(Comm(u2,nu^15));
infinity
gap> u2nu17 := Comm(u2,nu^17);
<bijective rcwa mapping of Z with modulus 81>
gap> cycs := ShortCycles(u2nu17,[-100..100],100);;
gap> List(cycs,Length);
[ 72, 72, 73, 72, 73, 72, 72, 73, 72, 72, 72, 73, 72, 72, 73, 72, 72, 
  73, 72, 72, 73, 72, 72 ]
gap> Lcm(last);
5256
gap> u2nu17^5256; # This element has indeed order 2^3*3^2*73 = 5256.
IdentityMapping( Integers )
gap> u2nu18 := Comm(u2,nu^18);
<bijective rcwa mapping of Z with modulus 81>
gap> cycs := ShortCycles(u2nu18,[-100..100],100);;
gap> List(cycs,Length);
[ 22, 22, 22, 21, 22, 22, 22, 21, 21, 22, 22, 21, 22, 21, 22, 22, 21, 
  22, 22, 21, 22, 22, 21 ]
gap> Lcm(last);
462
gap> u2nu18^462; # This is an element of order 2*3*7*11 = 462.
IdentityMapping( Integers )
gap> Order(Comm(u2,nu^20));
29
gap> Order(Comm(u2,nu^25));
9
gap> Order(Comm(u2,nu^30));
15
]]>
</Example>

Thus even this rather simple-minded approach reveals various different
orders of torsion elements, and the involved primes are also not all
very <Q>small</Q>.

</Section>

<!-- #################################################################### -->

<Section Label="sec:WildButFiniteCycles">
<Heading>A wild rcwa mapping which has only finite cycles</Heading>

Some wild rcwa mappings of <M>\mathbb{Z}</M> have only finite cycles. In this
section, a permutation is examined which can be shown to be such a mapping
and which is likely to be something like a <Q>minimal</Q> example. <P/>

Over <M>R</M> = GF(<M>q</M>)[<M>x</M>], the degree function gives rise
to a partition of <M>R</M> into finite sets which is left invariant by
suitable wild rcwa mappings. Over <M>R = \mathbb{Z}</M> the situation looks
different -- there is no such <Q>natural</Q> partition into finite sets
which can be fixed by a wild rcwa mapping.

<Example>
<![CDATA[
gap> kappa := RcwaMapping([[1,0,1],[1,0,1],[3,2,2],[1,-1,1],
>                          [2,0,1],[1,0,1],[3,2,2],[1,-1,1],
>                          [1,1,3],[1,0,1],[3,2,2],[2,-2,1]]);;
gap> SetName(kappa,"kappa");
gap> List([-5..5],k->Modulus(kappa^k));
[ 7776, 1296, 432, 72, 24, 1, 12, 72, 144, 864, 1728 ]
gap> Display(kappa);

Bijective rcwa mapping of Z with modulus 12

              n mod 12              |              n^kappa
------------------------------------+------------------------------------
   0  1  5  9                       | n
   2  6 10                          | (3n + 2)/2
   3  7                             | n - 1
   4                                | 2n
   8                                | (n + 1)/3
  11                                | 2n - 2

gap> List([-32..32],n->Length(Cycle(kappa,n)));
[ 4, 1, 4, 4, 7, 1, 10, 10, 1, 1, 4, 4, 7, 1, 10, 10, 4, 1, 7, 7, 1, 1, 
  7, 7, 4, 1, 4, 4, 2, 1, 1, 2, 1, 1, 4, 4, 4, 1, 7, 7, 4, 1, 7, 7, 1, 
  1, 10, 10, 7, 1, 4, 4, 7, 1, 10, 10, 1, 1, 4, 4, 4, 1, 13, 13, 7 ]
gap> List([2..14],k->Maximum(List([1..2^k],n->Length(Cycle(kappa,n)))));
[ 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 ]
gap> List([2..14],k->Length(Cycle(kappa,2^k-2)));
[ 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 ]
gap> Cycle(kappa,2^12-2);
[ 4094, 6142, 9214, 13822, 20734, 31102, 46654, 69982, 104974, 157462, 
  236194, 354292, 708584, 236195, 472388, 157463, 314924, 104975, 
  209948, 69983, 139964, 46655, 93308, 31103, 62204, 20735, 41468, 
  13823, 27644, 9215, 18428, 6143, 12284, 4095 ]
gap> last mod 12;
[ 2, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 4, 8, 11, 8, 11, 8, 11, 8, 
  11, 8, 11, 8, 11, 8, 11, 8, 11, 8, 11, 8, 11, 8, 3 ]
gap> lengthstats := Collected(List(ShortCycles(kappa,[1..12^4],100),
>                                  Length));
[ [ 1, 6912 ], [ 4, 1728 ], [ 7, 864 ], [ 10, 432 ], [ 13, 216 ], 
  [ 16, 108 ], [ 19, 54 ], [ 22, 27 ], [ 25, 13 ], [ 28, 7 ], [ 31, 3 ], 
  [ 34, 2 ], [ 37, 1 ], [ 40, 1 ] ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

We would like to determine a partition of&nbsp;<M>\mathbb{Z}</M> into unions
of cycles of equal length:

<Example>
<![CDATA[
gap> C := [Difference(Integers,MovedPoints(kappa))];; pow := [kappa^0];;
gap> rc := function(r,m) return ResidueClass(r,m); end;;
gap> for i in [1..3] do
>      Add(pow,kappa^i);
>      C[i+1] := Difference(rc(2,4),
>                  Union(Union(C{[1..i]}),
>                    Union(List([0..i],j->Intersection(
>                                           rc(2,4)^pow[j+1],
>                                           rc(2,4)^(pow[i-j+1]^-1))))));
>    od;
gap> C;
[ 1(4) U 0(12) U [ -2 ], 2(24) U 18(24), 6(48) U 38(48) U 10(72) U 58(72)
    , <union of 38 residue classes (mod 864)> ]
gap> List(C,S->Length(Cycle(kappa,S)));
[ 1, 4, 7, 10 ]
gap> Cycle(kappa,C[1]);
[ 1(4) U 0(12) U [ -2 ] ]
gap> Cycle(kappa,C[2]);
[ 2(24) U 18(24), 4(36) U 28(36), 8(72) U 56(72), 3(24) U 19(24) ]
gap> cycle7 := Cycle(kappa,C[3]);;
gap> for S in cycle7 do View(S); Print("\n"); od;
6(48) U 38(48) U 10(72) U 58(72)
10(72) U 58(72) U 16(108) U 88(108)
16(108) U 88(108) U 32(216) U 176(216)
11(72) U 59(72) U 32(216) U 176(216)
11(72) U 59(72) U 20(144) U 116(144)
7(48) U 39(48) U 20(144) U 116(144)
6(48) U 7(48) U 38(48) U 39(48)
gap> cycle10 := Cycle(kappa,C[4]);;
gap> for S in cycle10 do View(S); Print("\n"); od;
<union of 38 residue classes (mod 864)>
<union of 38 residue classes (mod 1296)>
<union of 12 residue classes (mod 648)>
<union of 12 residue classes (mod 648)>
<union of 22 residue classes (mod 1296)>
<union of 12 residue classes (mod 432)>
<union of 22 residue classes (mod 864)>
<union of 12 residue classes (mod 288)>
<union of 14 residue classes (mod 288)>
<union of 16 residue classes (mod 288)>
gap> List(cycle10,Density);
[ 19/432, 19/648, 1/54, 1/54, 11/648, 1/36, 11/432, 1/24, 7/144, 1/18 ]
gap> List(last,Float);
[ 0.0439815, 0.029321, 0.0185185, 0.0185185, 0.0169753, 0.0277778, 
  0.025463, 0.0416667, 0.0486111, 0.0555556 ]
gap> Sum(last2);
47/144
gap> Density(Union(cycle10));
47/432
]]>
</Example>

<Example>
<![CDATA[
gap> P := List(C,S->Union(Cycle(kappa,S)));;
gap> for S in P do View(S); Print("\n"); od;
1(4) U 0(12) U [ -2 ]
<union of 18 residue classes (mod 72)>
<union of 78 residue classes (mod 432)>
<union of 282 residue classes (mod 2592)>
gap> P2 := AsUnionOfFewClasses(P[2]);
[ 2(24), 3(24), 18(24), 19(24), 4(36), 28(36), 8(72), 56(72) ]
gap> Permutation(kappa,P2);
(1,5,7,2)(3,6,8,4)
gap> P3 := AsUnionOfFewClasses(P[3]);
[ 6(48), 7(48), 38(48), 39(48), 10(72), 11(72), 58(72), 59(72), 16(108), 
  88(108), 20(144), 116(144), 32(216), 176(216) ]
gap> Permutation(kappa,P3);
(1,5,9,13,6,11,2)(3,7,10,14,8,12,4)
gap> P4 := AsUnionOfFewClasses(P[4]);
[ 14(96), 15(96), 78(96), 79(96), 22(144), 23(144), 118(144), 119(144), 
  34(216), 35(216), 178(216), 179(216), 44(288), 236(288), 52(324), 
  268(324), 68(432), 356(432), 104(648), 536(648) ]
gap> Permutation(kappa,P4);
(1,5,9,15,19,10,17,6,13,2)(3,7,11,16,20,12,18,8,14,4)
gap> List(P,S->Set(List(Intersection([1..12^4],S),n->Length(Cycle(kappa,n)))));
[ [ 1 ], [ 4 ], [ 7 ], [ 10 ] ]
gap> Set(List(Intersection([1..12^4],Difference(Integers,Union(P))),
>             n->Length(Cycle(kappa,n))));
[ 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 ]
]]>
</Example>

Finally, the permutation <C>kappa</C> should be factored into involutions
(this time <Q>by hand</Q>, for purposes of illustration):

<Example>
<![CDATA[
gap> elm1 := kappa;
kappa
gap> Multpk(elm1,2,1)^elm1;
8(12)
gap> Multpk(elm1,2,-1)^elm1;
4(6)
gap> fact1 := ClassTransposition(4,6,8,12);;
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> elm2 := elm1/fact1;
<bijective rcwa mapping of Z with modulus 12>
gap> Display(elm2);

Bijective rcwa mapping of Z with modulus 12

              n mod 12              |                n^f
------------------------------------+------------------------------------
   0  1  4  5  9                    | n
   2  6 10                          | 3n + 2
   3  7 11                          | n - 1
   8                                | (n + 1)/3

gap> Multpk(elm2,3,1)^elm2;
8(12)
gap> Multpk(elm2,3,-1)^elm2;
3(4)
gap> fact2 := ClassTransposition(3,4,8,12);;
gap> elm3 := elm2/fact2;
<bijective rcwa mapping of Z with modulus 4>
gap> Display(elm3);

Bijective rcwa mapping of Z with modulus 4

              n mod 4               |                n^f
------------------------------------+------------------------------------
  0 1                               | n
  2                                 | n + 1
  3                                 | n - 1

gap> fact3 := ClassTransposition(2,4,3,4);;
gap> elm4 := elm3/fact3;
IdentityMapping( Integers )
gap> kappafacts := [ fact3, fact2, fact1 ];
[ ClassTransposition(2,4,3,4), ClassTransposition(3,4,8,12), 
  ClassTransposition(4,6,8,12) ]
gap> kappa = Product(kappafacts);
true
]]>
</Example>

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:AbelianModularGroup">
<Heading>An abelian rcwa group over a polynomial ring</Heading>

In this section, a wild rcwa group over GF(4)[<M>x</M>] should be
invstigated, which happens to be abelian. Of course in general, rcwa groups
also over this ring are usually far from being abelian (see below).
We start by defining this group:

<Example>
<![CDATA[
gap> x := Indeterminate(GF(4),1);; SetName(x,"x");
gap> R := PolynomialRing(GF(4),1);
GF(2^2)[x]
gap> e := One(GF(4));;
gap> p := x^2 + x + e;;    q := x^2 + e;;
gap> r := x^2 + x + Z(4);; s := x^2 + x + Z(4)^2;;
gap> cg := List( AllResidues(R,x^2), pol -> [ p, p * pol mod q, q ] );;
gap> ch := List( AllResidues(R,x^2), pol -> [ r, r * pol mod s, s ] );;
gap> g := RcwaMapping( R, q, cg );
<rcwa mapping of GF(2^2)[x] with modulus x^2+Z(2)^0>
gap> h := RcwaMapping( R, s, ch );
<rcwa mapping of GF(2^2)[x] with modulus x^2+x+Z(2^2)^2>
gap> List([g,h],Order);
[ infinity, infinity ]
gap> List([g,h],IsTame);
[ false, false ]
gap> G := Group(g,h);
<rcwa group over GF(2^2)[x] with 2 generators>
gap> IsAbelian(G);
true
]]>
</Example>

Now we compute the action of the group <C>G</C> on one of its orbits, and
make some statistics of the orbits of <C>G</C> containing polynomials of
degree less than&nbsp;4:

<Example>
<![CDATA[
gap> orb := Orbit(G,x^5);
[ x^5, x^5+x^4+x^2+Z(2)^0, x^5+x^3+x^2+Z(2^2)*x+Z(2)^0, x^5+x^3, 
  x^5+x^4+x^3+x^2+Z(2^2)^2*x+Z(2^2)^2, x^5+x, x^5+x^4+x^3, 
  x^5+x^2+Z(2^2)^2*x, x^5+x^4+x^2+x, x^5+x^3+x^2+Z(2^2)^2*x+Z(2)^0, 
  x^5+x^4+Z(2^2)*x+Z(2^2), x^5+x^3+x, x^5+x^4+x^3+x^2+Z(2^2)*x+Z(2^2), 
  x^5+x^4+x^3+x+Z(2)^0, x^5+x^2+Z(2^2)*x, x^5+x^4+Z(2^2)^2*x+Z(2^2)^2 ]
gap> H := Action(G,orb);
Group([ (1,2,4,7,6,9,12,14)(3,5,8,11,10,13,15,16), 
  (1,3,6,10)(2,5,9,13)(4,8,12,15)(7,11,14,16) ])
gap> IsAbelian(H); # check ...
true
gap> Exponent(H);
8
gap> Collected(List(ShortOrbits(G,AllResidues(R,x^4),100),Length));
[ [ 1, 4 ], [ 2, 6 ], [ 4, 12 ], [ 8, 24 ] ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

Changing the generators a little causes the group structure to change a lot:

<Example>
<![CDATA[
gap> cg[1][2] := cg[1][2] + (x^2 + e) * p * q;;
gap> ch[7][2] := ch[7][2] + x * r * s;;
gap> g := RcwaMapping( R, q, cg );; h := RcwaMapping( R, s, ch );;
gap> G := Group(g,h);
<rcwa group over GF(2^2)[x] with 2 generators>
gap> orb := Orbit(G,Zero(R));;
gap> Length(orb);
87
gap> Collected(List(orb,DegreeOfLaurentPolynomial));
[ [ 1, 2 ], [ 2, 4 ], [ 3, 16 ], [ 4, 64 ], [ infinity, 1 ] ]
gap> H := Action(G,orb);
<permutation group with 2 generators>
gap> IsNaturalAlternatingGroup(H);
true
gap> orb := Orbit(G,x^6);;
gap> Length(orb);
512
gap> H := Action(G,orb);
<permutation group with 2 generators>
gap> IsNaturalSymmetricGroup(H) or IsNaturalAlternatingGroup(H);
false
gap> blk := Blocks(H,[1..512]);;
gap> List(blk,Length);
[ 128, 128, 128, 128 ]
gap> Action(H,blk,OnSets);
Group([ (1,2)(3,4), (1,3)(2,4) ])
]]>
</Example>

Thus the modified group has a quotient isomorphic to the alternating group
of degree&nbsp;87, and a quotient isomorphic to some wreath product or a
subgroup thereof acting transitively, but not primitively on 512 points.

</Section>

<!-- #################################################################### -->

<Section Label="sec:TameByCommsOfWildPerms">
<Heading>
  A tame group generated by commutators of wild permutations
</Heading>

In this section, we have a look at 3 wild rcwa mappings whose commutators
generate tame groups:

<Example>
<![CDATA[
gap> a := RcwaMapping([[3,0,2],[3, 1,4],[3,0,2],[3,-1,4]]);;
gap> b := RcwaMapping([[3,0,2],[3,13,4],[3,0,2],[3,-1,4]]);;
gap> c := RcwaMapping([[3,0,2],[3, 1,4],[3,0,2],[3,11,4]]);;
gap> SetName(a,"a"); SetName(b,"b"); SetName(c,"c");
gap> List([a,b,c],IsTame);
[ false, false, false ]
gap> ab := Comm(a,b);; ac := Comm(a,c);; bc := Comm(b,c);;
gap> SetName(ab,"[a,b]"); SetName(ac,"[a,c]"); SetName(bc,"[b,c]");
gap> List([ab,ac,bc],Order);
[ 6, 6, 12 ]
]]>
</Example>

Now we would like to have a look at [<A>a</A>,<A>b</A>] ...

<Example>
<![CDATA[
gap> Display(ab);

Bijective rcwa mapping of Z with modulus 18, of order 6

              n mod 18              |              n^[a,b]
------------------------------------+------------------------------------
   0  2  3  8  9 11 12 17           | n
   1 10                             | 2n - 5
   4  7 13 16                       | n + 3
   5 14                             | 2n - 4
   6                                | (n + 2)/2
  15                                | (n - 5)/2

]]>
</Example>

... form the group generated by [<A>a</A>,<A>b</A>]
and [<A>a</A>,<A>c</A>] and compute its action on one of its orbits:

<Example>
<![CDATA[
gap> G := Group(ab,ac);
<rcwa group over Z with 2 generators>
gap> orb := Orbit(G,1);
[ -15, -12, -7, -6, -5, -4, -3, -2, -1, 1 ]
gap> H := Action(G,orb);
Group([ (2,5,8,10,7,6), (1,3,6,9,4,5) ])
gap> Size(H);
3628800
gap> Size(G); # G acts faithfully on orb.
3628800
]]>
</Example>

Hence the group <A>G</A> is isomorphic to the symmetric group on
10&nbsp;points and acts faithfully on the orbit containing&nbsp;1.

Another question is which groups arise if we take as generators
either <A>ab</A>, <A>ac</A> or <A>bc</A> and the involution
which maps any integer to its additive inverse:

<Example>
<![CDATA[
gap> t := ClassReflection(0,1);;
gap> Display(t);
Bijective rcwa mapping of Z: n -> -n
gap> G := Group(ab,t);
<rcwa group over Z with 2 generators>
gap> Size(G);
7257600
gap> phi := IsomorphismPermGroup(G);
[ [a,b], ClassReflection(0,1) ] ->
[ (1,36,12,27,9,15)(2,34,10,25,7,13)(3,35,11,26,8,14), 
  (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(20,21)(22,
    36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30) ]
gap> StructureDescription(Image(phi));
"C2 x S10"
]]>
</Example>

Thus the group generated by <A>ab</A> and <A>t</A> is isomorphic to 
<Alt Only="LaTeX"><M>{\rm C}_2 \times {\rm S}_{10}</M></Alt>
<Alt Not="LaTeX"><M>C_2 x S_10</M></Alt>.
The next group is an extension of a perfect group of order&nbsp;960:

<Example>
<![CDATA[
gap> G := Group(ac,t);;
gap> Size(G);
3840
gap> H := Image(IsomorphismPermGroup(G));;
gap> P := DerivedSubgroup(H);;
gap> Size(P);
960
gap> IsPerfect(P);
true
gap> PerfectGroup(PerfectIdentification(P));
A5 2^4'
]]>
</Example>

The last group is infinite:

<Example>
<![CDATA[
gap> G := Group(bc,t);;
gap> Size(G);
infinity
gap> Order(bc*t);
infinity
gap> Modulus(G);
18
gap> RespectedPartition(G);
[ 1(9), 2(9), 4(9), 5(9), 7(9), 8(9), 0(18), 3(18), 6(18), 9(18), 
  12(18), 15(18) ]
gap> ActionOnRespectedPartition(G);
Group([ (1,5,8,2,4,12)(3,9,6,11), (1,6)(2,5)(3,4)(8,12)(9,11) ])
gap> StructureDescription(last);
"S10"
gap> RankOfKernelOfActionOnRespectedPartition(G);
9
]]>
</Example>

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:CheckingForSolvability">
<Heading>Checking for solvability</Heading>

Is the group generated by the permutations <A>a</A> and <A>b</A> from the
last paragraph solvable? <P/>

This group is wild. Presently there is no general method available for
testing wild rcwa groups for solvability. But nevertheless, for the given
group we can obtain a negative answer to this question. The idea is to
find a subgroup&nbsp;<A>U</A> which acts on a finite set&nbsp;<A>S</A> of
integers, and which induces on&nbsp;<A>S</A> a non-solvable finite
permutation group:

<Example>
<![CDATA[
gap> a := RcwaMapping([[3,0,2],[3, 1,4],[3,0,2],[3,-1,4]]);; SetName(a,"a");
gap> b := RcwaMapping([[3,0,2],[3,13,4],[3,0,2],[3,-1,4]]);; SetName(b,"b");
gap> G := Group(a,b);;
gap> ShortOrbits(Group(Comm(a,b)),[-10..10],100);
[ [ -10 ], [ -9 ], [ -30, -21, -14, -13, -11, -8 ], [ -7 ], [ -6 ], 
  [ -12, -5, -4, -3, -2, 1 ], [ -1 ], [ 0 ], [ 2 ], [ 3 ], 
  [ 4, 5, 6, 7, 10, 15 ], [ 8 ], [ 9 ] ]
gap> S := [ 4, 5, 6, 7, 10, 15 ];;
gap> Cycle(Comm(a,b),4);
[ 4, 7, 10, 15, 5, 6 ]
gap> elm := RepresentativeAction(G,S,Permuted(S,(1,4)),OnTuples);
<bijective rcwa mapping of Z with modulus 81>
gap> List(S,n->n^elm);
[ 7, 5, 6, 4, 10, 15 ]
gap> U := Group(Comm(a,b),elm);
<rcwa group over Z with 2 generators>
gap> Action(U,S);
Group([ (1,4,5,6,2,3), (1,4) ])
gap> IsNaturalSymmetricGroup(last);
true
]]>
</Example>

Thus the subgroup <A>U</A> induces on <A>S</A> a natural symmetric group of
degree&nbsp;6. Therefore the group&nbsp;<A>G</A> is not solvable, as claimed.
We conclude this example by factoring the group element <A>elm</A> into
generators:

<Example>
<![CDATA[
gap> F := FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> RepresentativeActionPreImage(G,S,Permuted(S,(1,4)),OnTuples,F);
a^-2*b^-2*a*b*a^-1*b*a*b^-2*a
gap> a^-2*b^-2*a*b*a^-1*b*a*b^-2*a = elm;
true
]]>
</Example>

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:LocalExample">
<Heading>Some examples over (semi)localizations of the integers</Heading>

We start with something one can observe when trying to <Q>transfer</Q> an
rcwa mapping from the ring of integers to one of its localizations:

<Example>
<![CDATA[
gap> a2 := LocalizedRcwaMapping(a,2);
<rcwa mapping of Z_( 2 ) with modulus 4>
gap> IsSurjective(a2); # As expected
true
gap> IsInjective(a2); # Why not??
false
gap> 0^a2;
0
gap> (1/3)^a2; # That's the reason!
0
]]>
</Example>

The above can also be explained easily by pointing out that the
modulus of the inverse of <C>a</C> is&nbsp;3, and that 3 is a unit
of&nbsp;<M>\mathbb{Z}_{(2)}</M>.
Moving to <M>\mathbb{Z}_{(2,3)}</M> solves this problem:

<Example>
<![CDATA[
gap> a23 := SemilocalizedRcwaMapping(a,[2,3]);
<rcwa mapping of Z_( 2, 3 ) with modulus 4>
gap> IsBijective(a23);
true
]]>
</Example>

We get additional finite cycles, e.g.:

<Example>
<![CDATA[
gap> List(ShortOrbits(Group(a23),[0..50]/5,50),orb->Cycle(a23,orb[1]));
[ [ 0 ], [ 1/5, 2/5, 3/5 ], 
  [ 4/5, 6/5, 9/5, 8/5, 12/5, 18/5, 27/5, 19/5, 13/5, 11/5, 7/5 ], 
  [ 1 ], [ 2, 3 ], [ 14/5, 21/5, 17/5 ], 
  [ 16/5, 24/5, 36/5, 54/5, 81/5, 62/5, 93/5, 71/5, 52/5, 78/5, 117/5, 
      89/5, 68/5, 102/5, 153/5, 116/5, 174/5, 261/5, 197/5, 149/5, 
      113/5, 86/5, 129/5, 98/5, 147/5, 109/5, 83/5, 61/5, 47/5, 34/5, 
      51/5, 37/5, 29/5, 23/5 ], [ 4, 6, 9, 7, 5 ] ]
gap> List(last,Length);
[ 1, 3, 11, 1, 2, 3, 34, 5 ]
gap> List(ShortOrbits(Group(a23),[0..50]/7,50),orb->Cycle(a23,orb[1]));
[ [ 0 ], [ -1/7, 1/7 ], [ 2/7, 3/7, 4/7, 6/7, 9/7, 5/7 ], [ 1 ], 
  [ 2, 3 ], [ 4, 6, 9, 7, 5 ] ]
gap> List(last,Length);
[ 1, 2, 6, 1, 2, 5 ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

But the group structure remains invariant under the <Q>transfer</Q>
of a group with prime set <M>\{2,3\}</M> from <M>\mathbb{Z}</M>
to&nbsp;<M>\mathbb{Z}_{(2,3)}</M>:

<Example>
<![CDATA[
gap> b23 := SemilocalizedRcwaMapping(b,[2,3]);;
gap> c23 := SemilocalizedRcwaMapping(c,[2,3]);;
gap> ab23 := Comm(a23,b23);
<rcwa mapping of Z_( 2, 3 ) with modulus 18>
gap> ac23 := Comm(a23,c23);
<rcwa mapping of Z_( 2, 3 ) with modulus 18>
gap> G := Group(ab23,ac23);
<rcwa group over Z_( 2, 3 ) with 2 generators>
gap> S := Intersection(Enumerator(Rationals){[1..128]},Z_pi([2,3]));
[ -10, -9, -8, -7, -6, -5, -4, -3, -2, -9/5, -8/5, -10/7, -7/5, -9/7, 
  -6/5, -8/7, -1, -6/7, -4/5, -5/7, -3/5, -4/7, -3/7, -2/5, -2/7, -1/5, 
  -1/7, 0, 1/11, 1/7, 1/5, 2/7, 2/5, 3/7, 4/7, 3/5, 5/7, 4/5, 6/7, 1, 
  8/7, 6/5, 9/7, 7/5, 10/7, 8/5, 9/5, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> orbs := ShortOrbits(G,S,50);;
gap> List(orbs,Length);
[ 10, 1, 10, 1, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 
  10, 1, 1, 10, 1 ]
gap> ForAll(orbs,orb->IsNaturalSymmetricGroup(Action(G,orb)));
true
]]>
</Example>

<Q>Transferring</Q> a non-invertible rcwa mapping from the ring of integers
to some of its (semi)localizations can also turn it into an invertible one:

<Example>
<![CDATA[
gap> v := RcwaMapping([[6,0,1],[1,-7,2],[6,0,1],[1,-1,1],
>                      [6,0,1],[1, 1,2],[6,0,1],[1,-1,1]]);;
gap> SetName(v,"v");
gap> Display(v);

Rcwa mapping of Z with modulus 8

              n mod 8               |                n^v
------------------------------------+------------------------------------
  0 2 4 6                           | 6n
  1                                 | (n - 7)/2
  3 7                               | n - 1
  5                                 | (n + 1)/2

]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> IsInjective(v);
true
gap> IsSurjective(v);
false
gap> Image(v);
Z \ 4(12) U 8(12)
gap> Difference(Integers,last);
4(12) U 8(12)
gap> v2 := LocalizedRcwaMapping(v,2);
<rcwa mapping of Z_( 2 ) with modulus 8>
gap> IsBijective(v2);
true
gap> Display(v2^-1);

Bijective rcwa mapping of Z_( 2 ) with modulus 4

              n mod 4               |                n^f
------------------------------------+------------------------------------
  0                                 | 1/3 n / 2
  1                                 | 2 n + 7
  2                                 | n + 1
  3                                 | 2 n - 1

gap> S := ResidueClass(Z_pi(2),2,0);; l := [S];;
gap> for i in [1..10] do Add(l,l[Length(l)]^v2); od;
gap> l; # Visibly v2 is wild ...
[ 0(2), 0(4), 0(8), 0(16), 0(32), 0(64), 0(128), 0(256), 0(512),
  0(1024), 0(2048) ]
gap> w2 := RcwaMapping(Z_pi(2),[[1,0,2],[2,-1,1],[1,1,1],[2,-1,1]]);;
gap> v2w2 := Comm(v2,w2);; SetName(v2w2,"[v2,w2]"); v2w2^-1;;
gap> Display(v2w2);

Bijective rcwa mapping of Z_( 2 ) with modulus 8

              n mod 8               |             n^[v2,w2]
------------------------------------+------------------------------------
  0 3 4 7                           | n
  1                                 | n + 4
  2 6                               | 3 n
  5                                 | n - 4

]]>
</Example>

Again, viewed as an rcwa mapping of the integers the commutator given at
the end of the example would not be surjective.

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:Twisting257CyclesToModulus32">
<Heading>
  Twisting 257-cycles into an rcwa mapping with modulus 32
</Heading>

We define an rcwa mapping <A>x</A> of order&nbsp;257 with modulus&nbsp;32.
The easiest way to construct such a mapping is to prescribe a transition
graph and then to assign suitable affine mappings to its vertices.

<Example>
<![CDATA[
gap> x := RcwaMapping(
>           [[ 16,  2,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
>            [  1, 16,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
>            [  1, 16,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
>            [  1, 16,  1], [ 16, 18,  1], [  1, 16,  1], [ 16, 18,  1],
>            [  1,  0, 16], [ 16, 18,  1], [  1,-14,  1], [ 16, 18,  1],
>            [  1,-14,  1], [ 16, 18,  1], [  1,-14,  1], [ 16, 18,  1],
>            [  1,-14,  1], [ 16, 18,  1], [  1,-14,  1], [ 16, 18,  1],
>            [  1,-14,  1], [ 16, 18,  1], [  1,-14,  1], [  1,-31,  1]]);;
gap> SetName(x,"x"); Order(x);; Display(x);

Bijective rcwa mapping of Z with modulus 32, of order 257

              n mod 32              |                n^x
------------------------------------+------------------------------------
   0                                | 16n + 2
   1  3  5  7  9 11 13 15 17 19 21  |
  23 25 27 29                       | 16n + 18
   2  4  6  8 10 12 14              | n + 16
  16                                | n/16
  18 20 22 24 26 28 30              | n - 14
  31                                | n - 31

gap> Cycle(x,[1],0);
[ 0, 2, 18, 4, 20, 6, 22, 8, 24, 10, 26, 12, 28, 14, 30, 16, 1, 34, 50, 
  36, 52, 38, 54, 40, 56, 42, 58, 44, 60, 46, 62, 48, 3, 66, 82, 68, 84, 
  70, 86, 72, 88, 74, 90, 76, 92, 78, 94, 80, 5, 98, 114, 100, 116, 102, 
  118, 104, 120, 106, 122, 108, 124, 110, 126, 112, 7, 130, 146, 132, 
  148, 134, 150, 136, 152, 138, 154, 140, 156, 142, 158, 144, 9, 162, 
  178, 164, 180, 166, 182, 168, 184, 170, 186, 172, 188, 174, 190, 176, 
  11, 194, 210, 196, 212, 198, 214, 200, 216, 202, 218, 204, 220, 206, 
  222, 208, 13, 226, 242, 228, 244, 230, 246, 232, 248, 234, 250, 236, 
  252, 238, 254, 240, 15, 258, 274, 260, 276, 262, 278, 264, 280, 266, 
  282, 268, 284, 270, 286, 272, 17, 290, 306, 292, 308, 294, 310, 296, 
  312, 298, 314, 300, 316, 302, 318, 304, 19, 322, 338, 324, 340, 326, 
  342, 328, 344, 330, 346, 332, 348, 334, 350, 336, 21, 354, 370, 356, 
  372, 358, 374, 360, 376, 362, 378, 364, 380, 366, 382, 368, 23, 386, 
  402, 388, 404, 390, 406, 392, 408, 394, 410, 396, 412, 398, 414, 400, 
  25, 418, 434, 420, 436, 422, 438, 424, 440, 426, 442, 428, 444, 430, 
  446, 432, 27, 450, 466, 452, 468, 454, 470, 456, 472, 458, 474, 460, 
  476, 462, 478, 464, 29, 482, 498, 484, 500, 486, 502, 488, 504, 490, 
  506, 492, 508, 494, 510, 496, 31 ]
gap> Length(last);
257
]]>
</Example>

</Section>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<!-- #################################################################### -->

<Section Label="sec:ModuliOfPowers">
<Heading> The behaviour of the moduli of powers </Heading>

In this section some examples are given, which illustrate how different the
series of the moduli of powers of a given rcwa mapping of the integers
can look like.

<Example>
<![CDATA[
gap> List([0..4],i->Modulus(a^i));
[ 1, 4, 16, 64, 256 ]
gap> List([0..6],i->Modulus(ab^i));
[ 1, 18, 18, 18, 18, 18, 1 ]
gap> g:=RcwaMapping([[2,2,1],[1, 4,1],[1,0,2],[2,2,1],[1,-4,1],[1,-2,1]]);;
gap> h:=RcwaMapping([[2,2,1],[1,-2,1],[1,0,2],[2,2,1],[1,-1,1],[1, 1,1]]);;
gap> List([0..7],i->Modulus(g^i));
[ 1, 6, 12, 12, 12, 12, 6, 1 ]
gap> List([1..18],i->Modulus((g^3*h)^i));
[ 12, 6, 12, 12, 12, 6, 12, 6, 12, 12, 12, 6, 12, 6, 12, 12, 12, 6 ]
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> List([0..3],i->Modulus(u^i));
[ 1, 5, 25, 125 ]
gap> v6 := RcwaMapping([[-1,2,1],[1,-1,1],[1,-1,1]]);;
gap> List([0..6],i->Modulus(v6^i));
[ 1, 3, 3, 3, 3, 3, 1 ]
gap> w8 := RcwaMapping([[-1,3,1],[1,-1,1],[1,-1,1],[1,-1,1]]);;
gap> List([0..8],i->Modulus(w8^i));
[ 1, 4, 4, 4, 4, 4, 4, 4, 1 ]
gap> z := RcwaMapping([[2,  1, 1],[1,  1,1],[2, -1,1],[2, -2,1],
>                      [1,  6, 2],[1,  1,1],[1, -6,2],[2,  5,1],
>                      [1,  6, 2],[1,  1,1],[1,  1,1],[2, -5,1],
>                      [1,  0, 1],[1, -4,1],[1,  0,1],[2,-10,1]]);;
gap> SetName(z,"z");
gap> IsBijective(z);
true
gap> Display(z);

Bijective rcwa mapping of Z with modulus 16

              n mod 16              |                n^z
------------------------------------+------------------------------------
   0                                | 2n + 1
   1  5  9 10                       | n + 1
   2                                | 2n - 1
   3                                | 2n - 2
   4  8                             | (n + 6)/2
   6                                | (n - 6)/2
   7                                | 2n + 5
  11                                | 2n - 5
  12 14                             | n
  13                                | n - 4
  15                                | 2n - 10

]]>
</Example>

<Example>
<![CDATA[
gap> List([0..25],i->Modulus(z^i));
[ 1, 16, 32, 64, 64, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 
  256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024 ]
gap> e1 := RcwaMapping([[1,4,1],[2,0,1],[1,0,2],[2,0,1]]);;
gap> e2 := RcwaMapping([[1,4,1],[2,0,1],[1,0,2],[1,0,1],
>                       [1,4,1],[2,0,1],[1,0,1],[1,0,1]]);;
gap> List([e1,e2],Order);
[ infinity, infinity ]
gap> List([1..20],i->Modulus(e1^i));
[ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ]
gap> List([1..20],i->Modulus(e2^i));
[ 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4 ]
gap> SetName(e1,"e1"); SetName(e2,"e2"); Display(e2);

Bijective rcwa mapping of Z with modulus 8, of order infinity

              n mod 8               |                n^e2
------------------------------------+------------------------------------
  0 4                               | n + 4
  1 5                               | 2n
  2                                 | n/2
  3 6 7                             | n

gap> e2^2 = Restriction(RcwaMapping([[1,2,1]]),RcwaMapping([[4,0,1]]));
true
]]>
</Example>

</Section>

<!-- #################################################################### -->

<Section Label="sec:CollatzImagesAndPreImages">
<Heading> Images and preimages under the Collatz mapping </Heading>

We have a look at the images of the residue class 1(2)
under powers of the Collatz mapping.

<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);;
gap> S0 := ResidueClass(Integers,2,1);;
gap> S1 := S0^T;
2(3)
gap> S2 := S1^T;
1(3) U 8(9)
gap> S3 := S2^T;
2(3) U 4(9)
gap> S4 := S3^T;
Z \ 0(3) U 5(9)
gap> S5 := S4^T;
Z \ 0(3) U 7(9)
gap> S6 := S5^T;
Z \ 0(3)
gap> S7 := S6^T;
Z \ 0(3)
]]>
</Example>

Thus the image gets stable after applying the mapping <M>T</M> for
the 6th time. Hence <M>T^6</M> maps the residue class 1(2) surjectively
onto the union of the residue classes 1(3) and 2(3), which <M>T</M>
stabilizes setwise.
Now we would like to determine the preimages of 1(3) and 2(3) in 1(2)
under <M>T^6</M>. The residue class 1(2) has to be the disjoint union of
these sets.

<Example>
<![CDATA[
gap> U := Intersection(PreImage(T^6,ResidueClass(Integers,3,1)),S0);
<union of 11 residue classes (mod 64)>
gap> V := Intersection(PreImage(T^6,ResidueClass(Integers,3,2)),S0);
<union of 21 residue classes (mod 64)>
gap> AsUnionOfFewClasses(U);
[ 1(64), 5(64), 7(64), 9(64), 21(64), 23(64), 29(64), 31(64), 49(64), 
  51(64), 59(64) ]
gap> AsUnionOfFewClasses(V);
[ 3(32), 11(32), 13(32), 15(32), 25(32), 17(64), 19(64), 27(64), 33(64), 
  37(64), 39(64), 41(64), 53(64), 55(64), 61(64), 63(64) ]
gap> Union(U,V) = S0 and Intersection(U,V) = [];  # consistency check
true
]]>
</Example>

The images of the residue class 0(3) under powers of&nbsp;<M>T</M> look
as follows:

<Example>
<![CDATA[
gap> S0 := ResidueClass(Integers,3,0);
0(3)
gap> S1 := S0^T;
0(3) U 5(9)
gap> S2 := S1^T;
0(3) U 5(9) U 7(9) U 8(27)
gap> S3 := S2^T;
<union of 20 residue classes (mod 27)>
gap> S4 := S3^T;
<union of 73 residue classes (mod 81)>
gap> S5 := S4^T;
Z \ 10(81) U 37(81)
gap> S6 := S5^T;
Integers
gap> S7 := S6^T;
Integers
]]>
</Example>

Thus every integer is the image of a multiple of&nbsp;3
under&nbsp;<M>T^6</M>. This means that it would be sufficient to prove the
<M>3n+1</M> Conjecture for multiples of&nbsp;3.
We can obtain the corresponding result for multiples of&nbsp;5 as follows:

<Example>
<![CDATA[
gap> S := [ResidueClass(Integers,5,0)];
[ 0(5) ]
gap> for i in [1..12] do Add(S,S[i]^T); od;
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> for s in S do View(s); Print("\n"); od;
0(5)
0(5) U 8(15)
0(5) U 4(15) U 8(15)
0(5) U 2(15) U 4(15) U 8(15) U 29(45)
<union of 73 residue classes (mod 135)>
<union of 244 residue classes (mod 405)>
<union of 784 residue classes (mod 1215)>
<union of 824 residue classes (mod 1215)>
<union of 2593 residue classes (mod 3645)>
<union of 2647 residue classes (mod 3645)>
<union of 2665 residue classes (mod 3645)>
<union of 2671 residue classes (mod 3645)>
1(3) U 2(3) U 0(15)
gap> Union(S[13],ResidueClass(Integers,3,0));
Integers
gap>  List(S,Si->Float(Density(Si)));
[ 0.2, 0.266667, 0.333333, 0.422222, 0.540741, 0.602469, 0.645267, 
  0.678189, 0.711385, 0.7262, 0.731139, 0.732785, 0.733333 ]
]]>
</Example>

</Section>

<!-- #################################################################### -->

<Section Label="sec:FourTransitiveGroup">
<Heading>
  A group which acts 4-transitively on the positive integers
</Heading>

In this section, we would like to show that the group <M>G</M> generated
by the two wild mappings

<Example>
<![CDATA[
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);;
]]>
</Example>

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:

<Example>
<![CDATA[
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)
]]>
</Example>

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

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
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
]]>
</Example>

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 <M>c_{r(m)} &gt; a_{r(m)}</M>. Then we easily see that
<M>(a_{r(m)}n+b_{r(m)})/c_{r(m)} &gt; n</M> implies
<M>n &lt; b_{r(m)}/(c_{r(m)}-a_{r(m)})</M>.
Thus we can restrict our considerations to integers
<M>n &lt; b_{\rm max}</M>, where <M>b_{\rm max}</M> is the largest
second entry of a coefficient triple of one of the group elements
in our list:

<Example>
<![CDATA[
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
]]>
</Example>

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

<Example>
<![CDATA[
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
]]>
</Example>

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&nbsp;to&nbsp;do</Q>.

The approach taken here is to show that the stabilizer of&nbsp;1
in&nbsp;<M>G</M> acts transitively on the set of positive integers greater
than&nbsp;1. We do this by similar means as used above for showing the
transitivity of the action of <M>G</M> on the positive integers.

We start by determining all products of at most 5 generators and their
inverses, which stabilize&nbsp;1 (taking at most 4-generator products would
not suffice!):

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
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
]]>
</Example>

The resulting products have various different not quite small moduli:

<Example>
<![CDATA[
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 ] ]
]]>
</Example>

Similar as before, we determine for any of the above mappings the
residue classes whose elements larger than the largest <M>b_{r(m)}</M>
- coefficient of the respective mapping are mapped to smaller integers:

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> decs := List(stabelm,DecreasingOn);;
gap> List(decs,Modulus);
[ 2, 3, 8, 25, 9, 9, 16, 100, 12, 50, 25, 75, 27, 81, 54, 81, 64, 400, 
  48, 200, 100, 72, 108, 400, 80, 200, 625, 25, 75, 45, 75, 75, 225, 81, 
  243, 81, 243, 144, 144, 81, 243, 216, 243, 243, 128, 1600, 64, 400, 
  400, 48, 144, 1600, 320, 400, 2500, 100, 100, 60, 96, 192, 108, 324, 
  144, 324, 972, 400, 400, 80, 80, 400, 2500, 100, 1250, 625, 625, 25, 
  75, 75, 75, 45, 135, 600, 120, 150, 1875, 75, 225, 135, 225, 225, 675, 
  243, 729, 243, 729, 108, 216, 243, 729, 162, 729, 2187, 144, 144, 144, 
  144, 81, 81, 243, 729, 1296, 324, 972, 243, 729, 2187, 1458, 2187 ]
gap> Lcm(last);
174960000
]]>
</Example>

Since the least common multiple of the moduli of these unions of residue
classes is as large as 174960000, directly forming their union and
checking whether it is equal to the set of integers would take relatively
much time and memory. However, starting with the set of integers and
subtracting the above sets one-by-one in a suitably chosen order is cheap:

<Example>
<![CDATA[
gap> SortParallel(decs,stabelm,
>      function(S1,S2)
>        return First([1..100],k->Factorial(k) mod Modulus(S1)=0)
>             < First([1..100],k->Factorial(k) mod Modulus(S2)=0);
>      end);
gap> S := Integers;;
gap> for i in [1..Length(decs)] do
>      S_old := S; S := Difference(S,decs[i]);
>      if S <> S_old then ViewObj(S); Print("\n"); fi;
>      if S = [] then maxind := i; break; fi;
>    od;
0(2)
2(6) U 4(6)
<union of 8 residue classes (mod 30)>
<union of 19 residue classes (mod 90)>
<union of 114 residue classes (mod 720)>
<union of 99 residue classes (mod 720)>
<union of 57 residue classes (mod 720)>
<union of 54 residue classes (mod 720)>
<union of 41 residue classes (mod 720)>
<union of 35 residue classes (mod 720)>
<union of 8 residue classes (mod 720)>
4(720) U 94(720) U 148(720) U 238(720)
<union of 24 residue classes (mod 5760)>
<union of 72 residue classes (mod 51840)>
<union of 48 residue classes (mod 51840)>
<union of 192 residue classes (mod 259200)>
<union of 168 residue classes (mod 259200)>
<union of 120 residue classes (mod 259200)>
<union of 96 residue classes (mod 259200)>
<union of 72 residue classes (mod 259200)>
<union of 60 residue classes (mod 259200)>
<union of 48 residue classes (mod 259200)>
<union of 24 residue classes (mod 259200)>
<union of 12 residue classes (mod 259200)>
<union of 24 residue classes (mod 777600)>
<union of 12 residue classes (mod 777600)>
111604(194400) U 14404(777600) U 208804(777600)
[  ]
]]>
</Example>

Similar as above, it remains to check that the <Q>small</Q> integers all
lie in the orbit containing&nbsp;2. Obviously, it is sufficient to check
that any integer greater than&nbsp;2 is mapped to a smaller one by some
suitably chosen element of the stabilizer under consideration:

<Example>
<![CDATA[
gap> Maximum(List(stabelm{[1..maxind]},
>                 f->Maximum(List(Coefficients(f),c->c[2]))));
6581
gap> Filtered([3..6581],n->Minimum(List(stabelm,elm->n^elm))>=n);
[ 4 ]
]]>
</Example>

We have to treat 4 separately:

<Example>
<![CDATA[
gap> 1^(u*a*u^2*a^-1*u);
1
gap> 4^(u*a*u^2*a^-1*u);
3
]]>
</Example>

Now we know that any positive integer greater than&nbsp;1 lies in the same
orbit under the action of the stabilizer of&nbsp;1 in&nbsp;<M>G</M>
as&nbsp;2, thus that this stabilizer acts transitively on
<M>\mathbb{N} \setminus \{1\}</M>. But this means that we have established
the 2-transitivity of the action of <M>G</M> on&nbsp;<M>\mathbb{N}</M>. <P/>

In the following, we essentially repeat the above steps to show that
this action is indeed 3-transitive:

<Example>
<![CDATA[
gap> tups := Concatenation(List([1..6],k->Tuples([1..4],k)));;
gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]],
>                                      l->PositionSublist(tup,l)=fail));;
gap> stab := [];;
gap> for tup in tups do
>      l := [1,2];
>      for i in tup do l := List(l,n->n^gens[i]); od;
>      if l = [1,2] then Add(stab,tup); fi;
>    od;
gap> Length(stab);
212
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i])));;
gap> decs := List(stabelm,DecreasingOn);;
gap> SortParallel(decs,stabelm,function(S1,S2)
>      return First([1..100],k->Factorial(k) mod Mod(S1)=0)
>           < First([1..100],k->Factorial(k) mod Mod(S2)=0); end);
gap> S := Integers;;
gap> for i in [1..Length(decs)] do
>      S_old := S; S := Difference(S,decs[i]);
>      if S <> S_old then ViewObj(S); Print("\n"); fi;
>      if S = [] then break; fi;
>    od;
Z \ 1(8) U 7(8)
<union of 151 residue classes (mod 240)>
<union of 208 residue classes (mod 720)>
<union of 51 residue classes (mod 720)>
<union of 45 residue classes (mod 720)>
<union of 39 residue classes (mod 720)>
<union of 33 residue classes (mod 720)>
<union of 23 residue classes (mod 720)>
<union of 19 residue classes (mod 720)>
<union of 17 residue classes (mod 720)>
<union of 16 residue classes (mod 720)>
<union of 14 residue classes (mod 720)>
<union of 8 residue classes (mod 720)>
<union of 7 residue classes (mod 720)>
238(360) U 4(720) U 148(720) U 454(720)
<union of 38 residue classes (mod 5760)>
<union of 37 residue classes (mod 5760)>
<union of 25 residue classes (mod 5760)>
<union of 21 residue classes (mod 5760)>
<union of 17 residue classes (mod 5760)>
<union of 16 residue classes (mod 5760)>
<union of 138 residue classes (mod 51840)>
<union of 48 residue classes (mod 51840)>
<union of 32 residue classes (mod 51840)>
<union of 20 residue classes (mod 51840)>
<union of 16 residue classes (mod 51840)>
<union of 68 residue classes (mod 259200)>
<union of 42 residue classes (mod 259200)>
<union of 32 residue classes (mod 259200)>
<union of 26 residue classes (mod 259200)>
<union of 25 residue classes (mod 259200)>
<union of 11 residue classes (mod 259200)>
<union of 10 residue classes (mod 259200)>
<union of 7 residue classes (mod 259200)>
13414(129600) U 2164(259200) U 66964(259200) U 228964(259200)
2164(259200) U 66964(259200) U 228964(259200)
[  ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f),c->c[2]))));
515816
gap> smallnum := [4..515816];;
gap> for i in [1..Length(stabelm)] do
>      smallnum := Filtered(smallnum,n->n^stabelm[i]>=n);
>    od;
gap> smallnum;
[  ]
]]>
</Example>

The same for 4-transitivity:

<Example>
<![CDATA[
gap> tups := Concatenation(List([1..8],k->Tuples([1..4],k)));;
gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]],
>                                      l->PositionSublist(tup,l)=fail));;
gap> stab := [];;
gap> for tup in tups do
>      l := [1,2,3];
>      for i in tup do l := List(l,n->n^gens[i]); od;
>      if l = [1,2,3] then Add(stab,tup); fi;
>    od;
gap> Length(stab);
528
gap> stabelm := [];;
gap> for i in [1..Length(stab)] do
>      elm := One(G);
>      for j in stab[i] do
>        if Modulus(elm) > 10000 then elm := fail; break; fi;
>        elm := elm * gens[j];
>      od;
>      if elm <> fail then Add(stabelm,elm); fi;
>    od;
gap> Length(stabelm);
334
gap> decs := List(stabelm,DecreasingOn);;
gap> SortParallel(decs,stabelm,
>      function(S1,S2)
>        return First([1..100],k->Factorial(k) mod Modulus(S1) = 0)
>             < First([1..100],k->Factorial(k) mod Modulus(S2) = 0);
>      end);
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> S := Integers;;
gap> for i in [1..Length(decs)] do
>      S_old := S; S := Difference(S,decs[i]);
>      if S <> S_old then ViewObj(S); Print("\n"); fi;
>      if S = [] then maxind := i; break; fi;
>    od;
Z \ 1(8) U 7(8)
<union of 46 residue classes (mod 72)>
<union of 20 residue classes (mod 72)>
4(18)
<union of 28 residue classes (mod 576)>
<union of 22 residue classes (mod 576)>
<union of 21 residue classes (mod 576)>
40(72) U 4(144) U 94(144) U 346(576) U 418(576)
<union of 16 residue classes (mod 576)>
<union of 15 residue classes (mod 576)>
4(144) U 94(144) U 346(576) U 418(576)
<union of 30 residue classes (mod 5184)>
<union of 26 residue classes (mod 5184)>
<union of 6 residue classes (mod 1296)>
<union of 504 residue classes (mod 129600)>
<union of 324 residue classes (mod 129600)>
<union of 282 residue classes (mod 129600)>
<union of 239 residue classes (mod 129600)>
<union of 218 residue classes (mod 129600)>
<union of 194 residue classes (mod 129600)>
<union of 154 residue classes (mod 129600)>
<union of 97 residue classes (mod 129600)>
<union of 85 residue classes (mod 129600)>
<union of 77 residue classes (mod 129600)>
<union of 67 residue classes (mod 129600)>
<union of 125 residue classes (mod 259200)>
<union of 108 residue classes (mod 259200)>
<union of 107 residue classes (mod 259200)>
<union of 101 residue classes (mod 259200)>
<union of 100 residue classes (mod 259200)>
<union of 84 residue classes (mod 259200)>
<union of 80 residue classes (mod 259200)>
<union of 76 residue classes (mod 259200)>
<union of 70 residue classes (mod 259200)>
<union of 66 residue classes (mod 259200)>
<union of 54 residue classes (mod 259200)>
<union of 53 residue classes (mod 259200)>
<union of 47 residue classes (mod 259200)>
<union of 43 residue classes (mod 259200)>
<union of 31 residue classes (mod 259200)>
<union of 24 residue classes (mod 259200)>
<union of 23 residue classes (mod 259200)>
<union of 13 residue classes (mod 259200)>
57406(129600) U 115006(129600) U 192676(259200) U 250276(259200)
57406(129600) U 192676(259200) U 250276(259200) U 374206(388800)
57406(129600) U 192676(259200) U 250276(259200)
250276(259200) U 57406(388800) U 316606(388800) U 451876(777600)
316606(388800) U 451876(777600) U 509476(777600) U 768676(777600)
<union of 18 residue classes (mod 3110400)>
451876(777600) U 509476(777600) U 705406(777600) U 768676(777600) U 
2649406(3110400)
451876(777600) U 705406(777600) U 768676(777600) U 2649406(3110400)
451876(777600) U 705406(777600) U 2649406(3110400)
705406(777600) U 2007076(3110400) U 2649406(3110400) U 2784676(3110400)
<union of 14 residue classes (mod 9331200)>
2260606(2332800) U 5759806(9331200) U 5895076(9331200) U 8227876(9331200)
4593406(6998400) U 15091006(27993600) U 17559076(27993600) U 24557476(
27993600)
<union of 14 residue classes (mod 83980800)>
18590206(20995200) U 24557476(83980800) U 45552676(83980800) U 71078206(
83980800)
[  ]
gap> Maximum(List(stabelm{[1..maxind]},
>                 f->Maximum(List(Coefficients(f),c->c[2]))));
58975
gap> smallnum := [5..58975];;
gap> for i in [1..maxind] do
>      smallnum := Filtered(smallnum,n->n^stabelm[i]>=n);
>    od;
gap> smallnum;
[  ]
]]>
</Example>

There is even some evidence that the degree of transitivity of the action of
<M>G</M> on the positive integers is higher than&nbsp;4:

<Example>
<![CDATA[
gap> phi := EpimorphismFromFreeGroup(G);
[ a, u ] -> [ a, u ]
gap> F := Source(phi);
<free group on the generators [ a, u ]>
gap> List([5..20],
>         n->RepresentativeActionPreImage(G,[1,2,3,4,5],
>                                           [1,2,3,4,n],OnTuples,F));
[ <identity ...>, a^-3*u^4*a*u^-2*a^2, 
  a^-2*u*a^-1*u*a^-1*u*a^-1*u*a^-1*u^-1*a, a^4*u^-2*a^-4, a^-1*u^-4*a, 
  u^2*a^-1*u^2*a^-1*u^-2, u^-2*a^-2*u^4, a^-1*u^2*a, a^-1*u^-6*a, 
  a^2*u^4*a^2*u^2, u^-4*a*u^-2*a^-3, a^-1*u^-2*a^-3*u^4*a^2, 
  a^3*u^2*a*u^2, a*u^-4*a*u^-4*a^-2, u^-2*a*u^2*a*u^-2, u^-4*a^2*u^2 ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

</Section>

<!-- #################################################################### -->

<Section Label="sec:ThreeButNotFourTransitiveGroup">
<Heading>
  A group which acts 3-transitively, but not 4-transitively on Z
</Heading>

In this section, we would like to show that the wild group <M>G</M> generated
by the two tame mappings <M>n \mapsto n + 1</M> and <M>\tau_{1(2),0(4)}</M>
acts 3-transitively, but not 4-transitively on the set of integers.

<Example>
<![CDATA[
gap> G := Group(ClassShift(0,1),ClassTransposition(1,2,0,4));
<rcwa group over Z with 2 generators>
gap> IsTame(G);
false
gap> (G.1^-2*G.2)^3*(G.1^2*G.2)^3; # G <> the free product C_infty * C_2.
IdentityMapping( Integers )
gap> Display(G);

Wild rcwa group over Z, generated by

[
Tame bijective rcwa mapping of Z: n -> n + 1

Bijective rcwa mapping of Z with modulus 4, of order 2

              n mod 4               |   n^ClassTransposition(1,2,0,4)
------------------------------------+------------------------------------
  0                                 | (n + 2)/2
  1 3                               | 2n - 2
  2                                 | n

]

]]>
</Example>

This group acts transitively on <M>\mathbb{Z}</M>, since already the cyclic
group generated by the first of the two generators does so. Next we have to
show that it acts 2-transitively. We essentially proceed as in the example
in the previous section, by checking that the stabilizer of&nbsp;0
acts transitively on <M>\mathbb{Z} \setminus \{0\}</M>.

<Example>
<![CDATA[
gap> gens := [ClassShift(0,1)^-1,ClassTransposition(1,2,0,4),ClassShift(0,1)];;
gap> tups := Concatenation(List([1..6],k->Tuples([-1,0,1],k)));;
gap> tups := Filtered(tups,tup->ForAll([[0,0],[-1,1],[1,-1]],
>                                      l->PositionSublist(tup,l)=fail));;
gap> Length(tups);
189
gap> stab := [];;
gap> for tup in tups do
>      n := 0;
>      for i in tup do n := n^gens[i+2]; od;
>      if n = 0 then Add(stab,tup); fi;
>    od;
gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i+2])));;
gap> Collected(List(stabelm,Modulus));
[ [ 4, 6 ], [ 8, 4 ], [ 16, 3 ] ]
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> decs := List(stabelm,DecreasingOn);
[ 0(4), 3(4), 0(4), 3(4), 2(4), 0(4), 4(8), 2(4), 2(4), 0(4), 1(4), 
  0(8), 3(8) ]
gap> Union(decs);
Integers
]]>
</Example>

Similar as in the previous section, it remains to check that the integers
with <Q>small</Q> absolute value all lie in the orbit containing&nbsp;1 under
the action of the stabilizer of&nbsp;0:

<Example>
<![CDATA[
gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f),c->AbsInt(c[2])))));
21
gap> S := [1];;
gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
gap> IsSubset(S,Difference([-21..21],[0])); # Not yet ..
false
gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
gap> IsSubset(S,Difference([-21..21],[0])); # ... but now!
true
]]>
</Example>

Now we have to check for 3-transitivity. Since we cannot find for every
residue class an element of the pointwise stabilizer of <M>\{0,1\}</M>
which properly divides its elements, we also have to take additions and
subtractions into consideration. Since the moduli of all of our stabilizer
elements are quite small, simply looking at sets of representatives is cheap:

<Example>
<![CDATA[
gap> tups := Concatenation(List([1..10],k->Tuples([-1,0,1],k)));;
gap> tups := Filtered(tups,tup->ForAll([[0,0],[-1,1],[1,-1]],
>                                      l->PositionSublist(tup,l)=fail));;
gap> Length(tups);
3069
gap> stab := [];;
gap> for tup in tups do
>      l := [0,1];
>      for i in tup do l := List(l,n->n^gens[i+2]); od;
>      if l = [0,1] then Add(stab,tup); fi;
>    od;
gap> Length(stab);
10
gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i+2])));;
gap> Maximum(List(stabelm,Modulus));
8
gap> Maximum(List(stabelm,
>                 f->Maximum(List(Coefficients(f),c->AbsInt(c[2])))));
8
]]>
</Example>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<Example>
<![CDATA[
gap> decsp := List(stabelm,elm->Filtered([9..16],n->n^elm<n));
[ [ 9, 13 ], [ 10, 12, 14, 16 ], [ 12, 16 ], [ 9, 13 ], [ 12, 16 ], 
  [ 9, 11, 13, 15 ], [ 9, 11, 13, 15 ], [ 12, 16 ], [ 12, 16 ], 
  [ 9, 11, 13, 15 ] ]
gap> Union(decsp);
[ 9, 10, 11, 12, 13, 14, 15, 16 ]
gap> decsm := List(stabelm,elm->Filtered([-16..-9],n->n^elm>n));
[ [ -15, -13, -11, -9 ], [ -16, -12 ], [ -16, -12 ], [ -15, -11 ], 
  [ -16, -14, -12, -10 ], [ -15, -11 ], [ -15, -11 ], 
  [ -16, -14, -12, -10 ], [ -16, -14, -12, -10 ], [ -15, -11 ] ]
gap> Union(decsm);
[ -16, -15, -14, -13, -12, -11, -10, -9 ]
gap> S := [2];;
gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
gap> IsSubset(S,Difference([-8..8],[0,1]));
true
]]>
</Example>

At this point we have established 3-transitivity.
It remains to check that the group <M>G</M> does not act 4-transitively.
We do this by checking that it is not transitive on 4-tuples (mod&nbsp;4).
Since <M>n</M>&nbsp;mod&nbsp;8 determines the image of <M>n</M> under
a generator of <M>G</M> (mod&nbsp;4), it suffices to compute (mod&nbsp;8):

<Example>
<![CDATA[
gap> orb := [[0,1,2,3]];;
gap> extend := function ()
>      local gen;
>      for gen in gens do
>        orb := Union(orb,List(orb,l->List(l,n->n^gen) mod 8));
>      od;
>    end;;
gap> repeat
>      old := ShallowCopy(orb);
>      extend(); Print(Length(orb),"\n");
>    until orb = old;
7
27
97
279
573
916
1185
1313
1341
1344
1344
gap> Length(Set(List(orb,l->l mod 4)));
120
gap> last < 4^4;
true
]]>
</Example>

This shows that <M>G</M> acts not 4-transitively on&nbsp;<M>\mathbb{Z}</M>.
The corresponding calculation for 3-tuples looks as follows:

<Example>
<![CDATA[
gap> orb := [[0,1,2]];;
gap> repeat
>      old := ShallowCopy(orb);
>      extend(); Print(Length(orb),"\n");
>    until orb = old;
7
27
84
207
363
459
503
512
512
gap> Length(Set(List(orb,l->l mod 4)));
64
gap> last = 4^3;
true
]]>
</Example>

Needless to say that the latter kind of argumentation is not suitable
for proving, but only for disproving <M>k</M>-transitivity. 

</Section>

<!-- #################################################################### -->

<Section Label="sec:GrigorchukGroups">
<Heading>
  Grigorchuk groups
</Heading>

In this section, we show how to construct finite quotients of the two
infinite periodic groups introduced by Rostislav Grigorchuk
in&nbsp;<Cite Key="Grigorchuk80"/> with the help of &RCWA;.

The first of these, nowadays known as <Q>Grigorchuk group</Q>, is
investigated in an example given on the &GAP; website -- see
<URL>http://www.gap-system.org/Doc/Examples/grigorchuk.html</URL>.

The &RCWA; package permits a simpler and more elegant construction
of the finite quotients of this group: The function <C>TopElement</C>
given on the mentioned webpage gets unnecessary, and the function
<C>SequenceElement</C> can be simplified as follows:

<Listing>
<![CDATA[
SequenceElement := function ( r, level )

  return Permutation(Product(Filtered([1..level-1],k->k mod 3 <> r),
                             k->ClassTransposition(    2^(k-1)-1,2^(k+1),
                                                   2^k+2^(k-1)-1,2^(k+1))),
                     [0..2^level-1]);
end;
]]>
</Listing>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

The actual constructors for the generators are modified as follows:

<Listing>
<![CDATA[
a := level -> Permutation(ClassTransposition(0,2,1,2),[0..2^level-1]);
b := level -> SequenceElement(0,level);
c := level -> SequenceElement(2,level);
d := level -> SequenceElement(1,level);
]]>
</Listing>

All computations given on the webpage can now be done just as with the
<Q>original</Q> construction of the quotients of the Grigorchuk group.
In the sequel, we construct finite quotients of the second group introduced
in&nbsp;<Cite Key="Grigorchuk80"/>:

<Example>
<![CDATA[
gap> FourCycle := RcwaMapping((4,5,6,7),[4..7]);
<bijective rcwa mapping of Z with modulus 4, of order 4>
gap> GrigorchukGroup2Generator := function ( level )
>      if level = 1 then return FourCycle; else
>        return   Restriction(FourCycle, RcwaMapping([[4,1,1]]))
>               * Restriction(FourCycle, RcwaMapping([[4,3,1]]))
>               * Restriction(GrigorchukGroup2Generator(level-1),
>                             RcwaMapping([[4,0,1]]));
>      fi;
>    end;;
gap> GrigorchukGroup2 := level -> Group(FourCycle,
>                                       GrigorchukGroup2Generator(level));;
]]>
</Example>

We can do similar things as shown in the example on the &GAP; webpage
for the <Q>first</Q> Grigorchuk group:

<Example>
<![CDATA[
gap> G := List([1..4],lev->GrigorchukGroup2(lev)); # The first 4 quotients.
[ <rcwa group over Z with 2 generators>, 
  <rcwa group over Z with 2 generators>, 
  <rcwa group over Z with 2 generators>, 
  <rcwa group over Z with 2 generators> ]
gap> H := List([1..4],lev->Action(G[lev],[0..4^lev-1])); # Isom. perm.-gps.
[ Group([ (1,2,3,4), (1,2,3,4) ]), 
  Group([ (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), 
      (1,5,9,13)(2,6,10,14)(4,8,12,16) ]), 
  <permutation group with 2 generators>, 
  <permutation group with 2 generators> ]
gap> List(H,Size);
[ 4, 1024, 4294967296, 1329227995784915872903807060280344576 ]
gap> List(last,n->Collected(Factors(n)));
[ [ [ 2, 2 ] ], [ [ 2, 10 ] ], [ [ 2, 32 ] ], [ [ 2, 120 ] ] ]
gap> List(H,NilpotencyClassOfGroup);      
[ 1, 6, 14, 40 ]
]]>
</Example>

</Section>

<!-- #################################################################### -->

<Section Label="sec:ForwardOrbit">
<Heading>
  Forward orbits of a monoid with 2 generators
</Heading>

The <M>3n+1</M> Conjecture asserts that the forward orbit of any positive
integer under the Collatz mapping <M>T</M> contains&nbsp;1. In contrast,
it seems likely that <Q>most</Q> trajectories of the two mappings
<Alt Only="LaTeX">
  <Display>
  <![CDATA[
    T_5^\pm: \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ \
    n \ \longmapsto \
    \begin{cases}
      \frac{n}{2}        & \text{if} \ n \ \text{even}, \\
      \frac{5n \pm 1}{2} & \text{if} \ n \ \text{odd}
    \end{cases}
  ]]>
  </Display>
</Alt>
<Alt Only="HTML"><![CDATA[<center>
  <img src = "t5pm.png" width = "372" height = "61"
       alt = "T5+/-: Z -> Z, n |-> (n/2 if n even, (5n+/-1)/2 if n odd)" />
</center>]]></Alt>
<Alt Only="Text"><Verb><![CDATA[
                                       /
                                      | n/2          if n even,
          T_5+/-:  Z -> Z,   n  |->  <
                                      | (5n +/- 1)/2 if n odd
                                       \
]]></Verb></Alt>
diverge.
However we can show by means of computation that the forward orbit of any
positive integer under the action of the monoid generated by the two
mappings <M>T_5^-</M> and&nbsp;<M>T_5^+</M> indeed contains&nbsp;1.
First of all, we enter the generators:

<Example>
<![CDATA[
gap> T5m := RcwaMapping([[1,0,2],[5,-1,2]]);;
gap> T5p := RcwaMapping([[1,0,2],[5, 1,2]]);;
]]>
</Example>

We look for a number <M>k</M> such that for any residue class <M>r(2^k)</M>
there is a product&nbsp;<M>f</M> of <M>k</M>&nbsp;mappings <M>T_5^\pm</M>
whose restriction to <M>r(2^k)</M> is given by <M>n \mapsto (an+b)/c</M>
where <M>c>a</M>:

<Example>
<![CDATA[
gap> k := 1;;
gap> repeat
>      maps := List(Tuples([T5m,T5p],k),Product);
>      decr := List(maps,DecreasingOn);
>      decreasable := Union(decr);
>      Print(k,": "); View(decreasable); Print("\n");
>      k := k + 1;
>    until decreasable = Integers;
1: 0(2)
2: 0(4)
3: Z \ 1(8) U 7(8)
4: 0(4) U 3(16) U 6(16) U 10(16) U 13(16)
5: Z \ 7(32) U 25(32)
6: <union of 48 residue classes (mod 64)>
7: Integers
]]>
</Example>

Thus <M>k=7</M> serves our purposes.
To be sure that for any positive integer <M>n</M> our monoid contains
a mapping <M>f</M> such that <M>n^f&lt;n</M>, we still need to check this
condition for <Q>small</Q>&nbsp;<M>n</M>. Since in case <M>c>a</M> we have
<M>(an+b)/c \geq n</M> if only if <M>n \leq b/(c-a)</M>, we only need to
check those <M>n</M> which are not larger than the largest coefficient
<M>b_{r(m)}</M> occuring in any of the products under consideration:

<Example>
<![CDATA[
gap> maxb := Maximum(List(maps,f->Maximum(List(Coefficients(f),t->t[2]))));
25999
gap> small := Filtered([1..maxb],n->ForAll(maps,f->n^f>=n));
[ 1, 7, 9, 11 ]
]]>
</Example>

This means that except of&nbsp;1, only for <M>n \in \{{7,9,11\}}</M> there
is no product of 7 mappings <M>T_5^\pm</M> which maps <M>n</M> to a smaller
integer. We check that also the forward orbits of these three integers
contain&nbsp;1 by successively computing preimages of&nbsp;1: 

<Example>
<![CDATA[
gap> S := [1];; k := 0;;
gap> repeat
>      S := Union(S,PreImage(T5m,S),PreImage(T5p,S));
>      k := k+1;
>    until IsSubset(S,small);
gap> k;
17
]]>
</Example>

</Section>

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<Section Label="sec:FreeGroup">
<Heading>
  Representations of the free group of rank 2
</Heading>

The free group of rank&nbsp;2 embeds into RCWA(<M>\mathbb{Z}</M>) --
in fact it embeds even in the subgroup which is generated by all class
transpositions.
An explicit embedding can be constructed by transferring the construction
of the so-called <Q>Schottky groups</Q> (cf.&nbsp;<Cite Key="LaHarpe00"/>,
page&nbsp;27) from PSL(2,<M>\mathbb{C}</M>) to RCWA(<M>\mathbb{Z}</M>)
(we use the notation from the cited book):

<Example>
<![CDATA[
gap> D := AllResidueClassesModulo(4);
[ 0(4), 1(4), 2(4), 3(4) ]
gap> gamma1 := RepresentativeAction(RCWA(Integers),
>                                   Difference(Integers,D[1]),D[2]);;
gap> gamma2 := RepresentativeAction(RCWA(Integers),
>                                   Difference(Integers,D[3]),D[4]);;
gap> F2 := Group(gamma1,gamma2);
<rcwa group over Z with 2 generators>
]]>
</Example>

We can do some checks:

<Example>
<![CDATA[
gap> X1 := Union(D{[1,2]});; X2 := Union(D{[3,4]});;
gap>     IsSubset(X1,X2^gamma1) and IsSubset(X1,X2^(gamma1^-1))
>    and IsSubset(X2,X1^gamma2) and IsSubset(X2,X1^(gamma2^-1));
true
]]>
</Example>

The generators are products of 3 class transpositions, each:

<Example>
<![CDATA[
gap> Factorization(gamma1);
[ ClassTransposition(0,2,1,2), ClassTransposition(3,4,5,8),
  ClassTransposition(0,2,1,8) ]
gap> Factorization(gamma2);
[ ClassTransposition(0,2,1,2), ClassTransposition(1,4,7,8),
  ClassTransposition(0,2,3,8) ]
]]>
</Example>

The above construction is used by <Ref Attr="IsomorphismRcwaGroup"
Label="for a group"/> to embed free groups of any rank&nbsp;<M>\geq 2</M>.
<P/>

We give another only slightly different representation of the free group
of rank&nbsp;2. We verify that it really is one by applying the
so-called <E>Table-Tennis Lemma</E> (see e.g.&nbsp;<Cite Key="LaHarpe00"/>,
Section&nbsp;II.B.) to the infinite cyclic groups generated by the two
generators and to the same two sets <C>X1</C> and <C>X2</C> as above:

<Example>
<![CDATA[
gap> r1 := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4);;
gap> r2 := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,3,4);;
gap> F2 := Group(r1^2,r2^2);; SetName(F2,"F_2");
gap> List(GeneratorsOfGroup(F2),IsTame);
[ false, false ]
gap>     IsSubset(X1,X2^F2.1) and IsSubset(X1,X2^(F2.1^-1))
>    and IsSubset(X2,X1^F2.2) and IsSubset(X2,X1^(F2.2^-1));
true
gap> [Sources(r1),Sinks(r1),Loops(r1)]; # compare with X1
[ [ 0(4) ], [ 1(4) ], [ 0(4), 1(4) ] ]
gap> [Sources(r2),Sinks(r2),Loops(r2)]; # compare with X2
[ [ 2(4) ], [ 3(4) ], [ 2(4), 3(4) ] ]
gap>    IsSubset(X1,Union(Sinks(r1))) and IsSubset(X1,Union(Sinks(r1^-1)))
>   and IsSubset(X2,Union(Sinks(r2))) and IsSubset(X2,Union(Sinks(r2^-1)));
true
gap> IsSubset(Union(Sinks(r1)),X2^F2.1) and
>    IsSubset(Union(Sinks(r1^-1)),X2^(F2.1^-1));
true
gap> IsSubset(Union(Sinks(r2)),X1^F2.2) and
>    IsSubset(Union(Sinks(r2^-1)),X1^(F2.2^-1));
true
]]>
</Example>

Drawing the transition graphs of <C>r1</C> and <C>r2</C> for modulus&nbsp;4
may help understanding what is actually done in this calculation.
It is easy to see that the group generated by <C>r1</C> and&nbsp;<C>r2</C>
is <E>not</E> free:

<Example>
<![CDATA[
gap> Order(r1/r2);
3
]]>
</Example>

</Section>

<!-- #################################################################### -->

<Section Label="sec:ModularGroup">
<Heading>
  Representations of the modular group PSL(2,Z)
</Heading>

The modular group PSL(2,<M>\mathbb{Z}</M>) embeds in the group generated by
all class transpositions as well. We give an embedding, and check that it
really is one by applying the Table Tennis Lemma as in the previous section:

<Example>
<![CDATA[
gap> PSL2Z := 
>      Group(ClassTransposition(0,3,1,3) * ClassTransposition(0,3,2,3),
>            ClassTransposition(1,3,0,6) * ClassTransposition(2,3,3,6));;
gap> List(GeneratorsOfGroup(PSL2Z),Order);
[ 3, 2 ]
]]>
</Example>

<Example>
<![CDATA[
gap> X1 := Difference(Integers,ResidueClass(0,3));
Z \ 0(3)
gap> X2 := ResidueClass(0,3);
0(3)
gap> IsSubset(X1,X2^PSL2Z.1) and IsSubset(X1,X2^(PSL2Z.1^2));
true
gap> IsSubset(X2,X1^PSL2Z.2);
true
]]>
</Example>

A slightly different representation of PSL(2,<M>\mathbb{Z}</M>) can be
obtained by using &RCWA;'s general method for <C>IsomorphismRcwaGroup</C>
for free products of finite groups:

<Example>
<![CDATA[
gap> Display(Image(IsomorphismRcwaGroup(FreeProduct(CyclicGroup(3),
>                                                   CyclicGroup(2)))));

Wild rcwa group over Z, generated by

[

Bijective rcwa mapping of Z with modulus 4

              n mod 4               |                n^f
------------------------------------+------------------------------------
  0                                 | n + 2
  1 3                               | 2n - 2
  2                                 | n/2


Bijective rcwa mapping of Z with modulus 2

              n mod 2               |                n^f
------------------------------------+------------------------------------
  0                                 | n + 1
  1                                 | n - 1

]

]]>
</Example>

<Alt Only="HTML">&nbsp;</Alt>

</Section>

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</Chapter>

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