GAP (RCWA) - Chapter 4: Examples

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### 4. Examples

This chapter discusses a number of "nice" 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.

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

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!

#### 4.1 Factoring Collatz' permutation of the integers

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

    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

This permutation has a few finite cycles:

    gap> ShortCycles(Collatz,[-50..50],50);
    [ [ -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 ] ]

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:

    gap> Length(Factorization(Collatz));
    212

Setting the Info level of `InfoRCWA` equal to 2 (simply issue `RCWAInfo(2);`) 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:

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

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

    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

The factors `PrimeSwitch(3)` are products of 6 class transpositions (cp. `PrimeSwitch` (2.2-4)). At the end of Section 4.6, a much smaller factorization task is performed "manually" for purposes of illustration.

#### 4.2 An rcwa mapping which seems to be contracting, but very slow

The iterates of an integer under the Collatz mapping T 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 3n+1 Conjecture has not been proven so far!):

    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 information on how to improve this guess.
    [ -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..40],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, 67, 5, 18, 10, 10, 15, 15, 15, 23, 7 ]
    gap> Maximum(List([1..1000],n->Length(Trajectory(T,n,S0))));
    113
    gap> Maximum(List([1..1000],n->Maximum(Trajectory(T,n,S0))));
    125252

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

    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.
    gap> Trajectory(f6,25,S0);
    [ 25, 21, 39, 72, 12, 2 ]
    gap> List([1..100],n->Length(Trajectory(f6,n,S0)));
    [ 2, 2, 3, 4, 2, 2, 3, 2, 2, 5, 7, 2, 8, 17, 3, 16, 2, 4, 17, 6, 5, 2, 5, 5,
      6, 2, 4, 2, 15, 2, 2, 3, 2, 5, 13, 3, 2, 3, 4, 2, 8, 4, 4, 2, 7, 19, 23517,
      3, 9, 3, 2, 18, 14, 2, 20, 23512, 14, 2, 6, 6, 2, 4, 19, 12, 23511, 8,
      23513, 10, 2, 13, 13, 3, 2, 23517, 7, 20, 7, 9, 9, 6, 12, 8, 6, 18, 14,
      23516, 31, 12, 23545, 4, 21, 19, 5, 2, 17, 17, 13, 19, 6, 23515 ]
    gap> Maximum(Trajectory(f6,47,S0));;
    736339177776247330443187705477107581873369010805146980871580925673774229545698\
    886054

Computing the trajectory of 3224 takes quite a while -- this trajectory ascends to about 3 * 10^2197, before it approaches the fixed point 2 after 19949562 steps.

When constructing the mapping `f6`, 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 `Modulus(f6)`th power of the denominator. In the example we have 5 * 7 * 11^3 = 46585 and 6^6 = 46656.

Although the trajectories of `T` are much shorter than those of `f6`, it seems likely that this does not make the problem of deciding whether the mapping `T` is contracting essentially easier -- even for mappings with much shorter trajectories than `T` 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 k such that applying the kth power of the respective mapping to any integer decreases its absolute value.

#### 4.3 Checking a result by P. Andaloro

In [A00], P. Andaloro has shown that proving that trajectories of integers n in 1(16) under the Collatz mapping always contain 1 would be sufficient to prove the 3n+1 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 T contains Z \ 0(3) is obviously enough. Thus we put S := 1(16), and successively unite the set S with its image under T:

    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

Further similar computations are shown in Section 4.13.

#### 4.4 Two examples by Matthews and Leigh

In [ML87], K. R. Matthews and G. M. Leigh have shown that two trajectories of the following (surjective, but not injective) mappings are acyclic (mod x) and divergent:

    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 ]

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 mod x and 1 mod x:

    gap> traj1modx := Trajectory(ML1,One(R),400,x);;
    gap> traj2modx := Trajectory(ML2,(x^3+x+1)*One(R),600,x);;
    gap> List(traj1modx{[1..200]},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,
      1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 ]
    gap> List(traj2modx{[1..200]},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,
      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 ]

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

    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 ]

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.

#### 4.5 Exploring the structure of a wild rcwa group

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:

    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

Now we would like to know which orders torsion elements of `G` can have -- taking a look at the above generators it seems to make sense to try commutators:

    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 ]

    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> orbs := ShortOrbits(Group(u2nu17),[-100..100],100);;
    gap> List(orbs,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> orbs := ShortOrbits(Group(u2nu18),[-100..100],100);;
    gap> List(orbs,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

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

#### 4.6 A wild rcwa mapping which has only finite cycles

Some wild rcwa mappings of Z 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 "minimal" example.

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

    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> lengthstatistics := Collected(List(ShortOrbits(Group(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 ] ]

We would like to determine a partition of Z into unions of cycles of equal length:

    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

    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 ]

Finally, the permutation `kappa` should be factored into involutions (this time "by hand", for purposes of illustration):

    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);;

    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

#### 4.7 An abelian rcwa group over a polynomial ring

In this section, a wild rcwa group over GF(4)[x] 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:

    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

Now we compute the action of the group `G` on one of its orbits, and make some statistics of the orbits of `G` containing polynomials of degree less than 4:

    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 ] ]

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

    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) ])

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

#### 4.8 A tame group generated by commutators of wild permutations

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

    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 ]

Now we would like to have a look at [a,b] ...

    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

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

    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

Hence the group G is isomorphic to the symmetric group on 10 points and acts faithfully on the orbit containing 1. Another question is which groups arise if we take as generators either ab, ac or bc and the involution which maps any integer to its additive inverse:

    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"

Thus the group generated by ab and t is isomorphic to C_2 x S_10. The next group is an extension of a perfect group of order 960:

    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'

The last group is infinite:

    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

#### 4.9 Checking for solvability

Is the group generated by the permutations a and b from the last paragraph solvable?

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 U which acts on a finite set S of integers, and which induces on S a non-solvable finite permutation group:

    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

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

    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

#### 4.10 Some examples over (semi)localizations of the integers

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

    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

The above can also be explained easily by pointing out that the modulus of the inverse of `a` is 3, and that 3 is a unit of Z_(2). Moving to Z_(2,3) solves this problem:

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

We get additional finite cycles, e.g.:

    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 ]

But the group structure remains invariant under the "transfer" of a group with prime set 2,3 from Z to Z_(2,3):

    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..200]},Z_pi([2,3]));
    [ -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -12/5, -11/5, -2, -9/5, -12/7,
      -8/5, -11/7, -10/7, -7/5, -9/7, -6/5, -8/7, -12/11, -1, -10/11, -6/7,
      -9/11, -4/5, -8/11, -5/7, -7/11, -3/5, -4/7, -6/11, -5/11, -3/7, -2/5,
      -4/11, -2/7, -3/11, -1/5, -2/11, -1/7, -1/11, 0, 1/13, 1/11, 1/7, 2/13,
      2/11, 1/5, 3/13, 3/11, 2/7, 4/13, 4/11, 5/13, 2/5, 3/7, 5/11, 6/13, 7/13,
      6/11, 4/7, 3/5, 8/13, 7/11, 9/13, 5/7, 8/11, 10/13, 4/5, 9/11, 11/13, 6/7,
      10/11, 12/13, 1, 12/11, 8/7, 13/11, 6/5, 9/7, 7/5, 10/7, 11/7, 8/5, 12/7,
      9/5, 2, 11/5, 12/5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ]
    gap> orbs := ShortOrbits(G,S,50);;
    gap> List(orbs,Length);
    [ 10, 10, 1, 10, 1, 10, 10, 10, 10, 10, 1, 10, 10, 10, 1, 10, 10, 10, 10, 10,
      10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1,
      10, 1, 10, 10, 10, 1, 1, 10, 1, 10 ]
    gap> ForAll(orbs,orb->IsNaturalSymmetricGroup(Action(G,orb)));
    true

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

    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

    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

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

#### 4.11 Twisting 257-cycles into an rcwa mapping with modulus 32

We define an rcwa mapping x of order 257 with modulus 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.

    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"); Display(x);

    Rcwa mapping of Z with modulus 32

                   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> Order(x);
    257
    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

#### 4.12 The behaviour of the moduli of powers

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.

    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..20],i->Modulus((g^3*h)^i));
    [ 12, 6, 12, 12, 12, 6, 12, 6, 12, 12, 12, 6, 12, 6, 12, 12, 12, 6, 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

    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");
    gap> 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

#### 4.13 Images and preimages under the Collatz mapping

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

    gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; 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)

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

    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

The images of the residue class 0(3) under powers of T look as follows:

    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

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

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

    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 ]

#### 4.14 A group which acts 4-transitively on the positive integers

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

    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);;

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:

    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)

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

    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

Finally, we have to deal with "small" 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_rm max, where b_rm max is the largest second entry of a coefficient triple of one of the group elements in our list:

    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

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:

    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

Checking 2-transitivity is computationally harder, and in the sequel we will omit some steps which are in practice needed to find out "what to do". 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!):

    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

The resulting products have various different not quite small moduli:

    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 ] ]

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:

    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

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:

    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)
    [  ]

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

    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 ]

We have to treat 4 separately:

    gap> 1^(u*a*u^2*a^-1*u);
    1
    gap> 4^(u*a*u^2*a^-1*u);
    3

Now we know that any positive integer greater than 1 lies in the same orbit under the action of the stabilizer of 1 in G as 2, thus that this stabilizer acts transitively on N \ 1. But this means that we have established the 2-transitivity of the action of G on N.

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

    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

    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 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 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)
    [  ]

    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;
    [  ]

The same for 4-transitivity:

    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);

    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;
    [  ]

There is even some evidence that the degree of transitivity of the action of G on the positive integers is higher than 4:

    gap> phi := EpimorphismFromFreeGroup(G);
    [ a, u ] -> [ a, u ]
    gap> F := Source(phi);
    <free group on the generators [ a, u ]>
    gap> words := 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 ]

#### 4.15 A group which acts 3-transitively, but not 4-transitively on Z

In this section, we would like to show that the wild group G generated by the two tame mappings n -> n + 1 and tau_1(2),0(4) acts 3-transitively, but not 4-transitively on the set of integers.

    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 is not 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

    ]

This group acts transitively on Z, 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 0 acts transitively on Z \ 0.

    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 ] ]

    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

Similar as in the previous section, it remains to check that the integers with "small" absolute value all lie in the orbit containing 1 under the action of the stabilizer of 0:

    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

Now we have to check for 3-transitivity. Since we cannot find for every residue class an element of the pointwise stabilizer of 0,1 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:

    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

    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

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

    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

This shows that G is not 4-transitive on Z. The corresponding calculation for 3-tuples looks as follows:

    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

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

#### 4.16 Grigorchuk groups

In this section, we show how to construct finite quotients of the two infinite periodic groups introduced by Rostislav Grigorchuk in [G80] with the help of RCWA. The first of these, nowadays known as "Grigorchuk group", is investigated in an example given on the GAP website -- see http://www.gap-system.org/Doc/Examples/grigorchuk.html. The RCWA package permits a simpler and more elegant construction of the finite quotients of this group: The function `TopElement` given on the mentioned webpage gets unnecessary, and the function `SequenceElement` can be simplified as follows:

    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;

The actual constructors for the generators are modified as follows:

    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);

All computations given on the webpage can now be done just as with the "original" construction of the quotients of the Grigorchuk group. In the sequel, we construct finite quotients of the second group introduced in [G80]:

    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));;

We can do similar things as shown in the example on the GAP webpage for the "first" Grigorchuk group:

    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])); # Isomorphic 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 ]

#### 4.17 Forward orbits of a monoid with 2 generators

The 3n+1 Conjecture asserts that the forward orbit of any positive integer under the Collatz mapping T contains 1. In contrast, it seems likely that "most" trajectories of the two mappings

                                           /
                                          | n/2          if n even,
              T_5+/-:  Z -> Z,   n  |->  <
                                          | (5n +/- 1)/2 if n odd
                                           \

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 T_5^- and T_5^+ indeed contains 1. First of all, we enter the generators:

    gap> T5m := RcwaMapping([[1,0,2],[5,-1,2]]);;
    gap> T5p := RcwaMapping([[1,0,2],[5, 1,2]]);;

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

    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

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

    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 ]

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

    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

#### 4.18 Representations of the free group of rank 2

The free group of rank 2 embeds in RCWA(Z) -- 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 "Schottky groups" (cp. [H00], page 27) from PSL(2,C) to RCWA(Z) (we use the notation from the cited book):

    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>

We can do some checks:

    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

The generators are products of 3 class transpositions, each:

    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) ]

The above construction is used by `IsomorphismRcwaGroup` (3.1-2) to embed free groups of any rank >= 2.

We give another only slightly different representation of the free group of rank 2. We verify that it really is one by applying the so-called Table-Tennis Lemma (see e.g. [H00], Section II.B.) to the infinite cyclic groups generated by the two generators and to the same two sets `X1` and `X2` as above:

    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

Drawing the transition graphs of `r1` and `r2` for modulus 4 may help understanding what is actually done in this calculation. It is easy to see that the group generated by `r1` and `r2` is not free:

    gap> Order(r1/r2);
    3

#### 4.19 Representations of the modular group PSL(2,Z)

The modular group PSL(2,Z) 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:

    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 ]

    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

A slightly different representation of PSL(2,Z) can be obtained by using RCWA's general method for `IsomorphismRcwaGroup` for free products of finite groups:

    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

    ]

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