[GAP Forum] RCWA 2.1

[Stefan Kohl]

Dear Forum,

This is to announce the release of RCWA 2.1.

The RCWA package provides methods for computing in certain infinite
permutation groups acting on the integers.

The class of groups which can be dealt with is closed under forming wreath
products with finite groups and with the infinite cyclic group (Z,+).
It includes free groups, all free products of finitely many finite groups
and certain divisible torsion groups.

The class transpositions - these are involutions which interchange two
disjoint residue classes - generate a countable simple group. This group
has subgroups of all mentioned types. Its class of isomorphism types of
subgroups is closed under the above operations. Therefore by a result of
Mihailova it has finitely generated subgroups with unsolvable membership
problem, and by a result of Baumslag, it has finitely generated subgroups
which are not finitely presented.

As usual, you can find the RCWA package at

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

Below I give some short examples illustrating a few recently added features.

One can enter a group by giving generators:

----------------------------------------------------------------------------
gap> G := Group(ClassTransposition(1,2,4,6),ClassShift(0,2));
<rcwa group over Z with 2 generators>
gap> StructureDescription(G);
"(Z x Z x Z x Z) . S4"
----------------------------------------------------------------------------

One can form direct products and wreath products:

----------------------------------------------------------------------------
gap> F2 := Image(IsomorphismRcwaGroup(FreeGroup(2)));
<wild rcwa group over Z with 2 generators>
gap> PSL2Z := Image(IsomorphismRcwaGroup(FreeProduct(CyclicGroup(2),
 >                                                    CyclicGroup(3))));
<wild rcwa group over Z with 2 generators>
gap> G := DirectProduct(WreathProduct(PSL2Z,Group(ClassShift(0,1))),
 >                       WreathProduct(F2,AlternatingGroup(4)),
 >                       Group(ClassShift(0,1),ClassTransposition(1,2,0,4)),
 >                       Group(ClassTransposition(1,2,0,6),ClassShift(1,3)));
<wild rcwa group over Z with 17 generators>
gap> StructureDescription(G);
"((C2 * C3) wr Z) x (F2 wr A4) x (<unknown> . Z) x ((Z x Z x Z) . D12)"
----------------------------------------------------------------------------

The method for forming wreath products with (Z,+) follows a suggestion
by Laurent Bartholdi.

Sections of the group on whose structure RCWA cannot obtain information
automatically are denoted by <unknown>. Such sections can have an
interesting group theoretic structure, which often can be explored in an
interactive session by a sequence of commands.

Wishing you fun and success using this package,

     Stefan Kohl

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http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/
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