[GAP Forum] RCWA 2.2

[Stefan Kohl]

Dear Forum,

This is to announce the release of RCWA 2.2.

The RCWA package can deal with groups of the following and many more types:

  - Finite groups.
  - Free groups of finite rank.
  - Free products of finitely many finite groups.
  - Certain divisible torsion groups.
  - Certain countable simple groups.
  - Direct products of the mentioned groups.
  - Wreath products of the mentioned groups with finite groups and with
    the infinite cyclic group (Z,+).

As usual, RCWA is available at

     http://www.gap-system.org/Packages/rcwa.html .

In analogy to Frank Lübeck's group recognition challenges for matrix groups
( http://www.math.rwth-aachen.de:8001/~Frank.Luebeck/data/MatrixChallenges/ ),
I list below a couple of `challenge groups' whose structure you might try
to determine:

G1 := Group( ClassTransposition(0,2,1,2),
              ClassTransposition(0,4,2,4) * ClassTransposition(1,2,0,4) );

G2 := Group( ClassTransposition(0,2,1,2) * ClassTransposition(0,2,1,4),
              ClassTransposition(3,8,7,8) * ClassTransposition(3,8,7,16) );

G3 := Group( ClassTransposition(0,4,3,4),
              ClassTransposition(0,6,3,6),
              ClassTransposition(1,4,0,6) );

G4 := ClosureGroup( G3,   ClassTransposition(0,4,3,4)
                         * ClassTransposition(6,12,9,12)
                         * ClassTransposition(0,6,9,12) );

G5 := ClosureGroup( G4, [ ClassTransposition(1,96,73,96),
                           ClassTransposition(1,144,73,144),
                           ClassTransposition(25,96,1,144) ] );

G6 := ClosureGroup( G5, [   ClassTransposition(17,48,41,48)
                           * ClassTransposition(41,48,65,96),
                             ClassTransposition(65,96,89,96)
                           * ClassTransposition(113,144,209,288)
                           * ClassTransposition(89,96,113,144) ] );

G7 := Group( mKnot(3), ClassShift(Integers) );

Just a few hints:

  - Some of the above groups are finitely-presented, others are not.
  - Most of them don't have finite-dimensional linear representations
    over a field.
  - Most of them don't appear in the list of types of groups given above.
  - Some of them can be recognized easily, others are quite complicate
    constructions.

Given disjoint residue classes r1(m1) and r2(m2) of Z,
ClassTransposition(r1,m1,r2,m2) in Sym(Z) is the unique involution which
interchanges r1(m1) and r2(m2), which is affine on r1(m1) and on r2(m2),
which maps nonnegative integers to nonnegative integers and which fixes
Z \ (r1(m1) U r2(m2)) pointwise. For example, ClassTransposition(0,2,1,2)
is given by n |-> n + (-1)^n.

Wishing you fun and success using this package,

     Stefan Kohl

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