43 Pc Groups

PcGroups are polycyclic groups that use the polycyclic presentation for element arithmetic. This presentation gives them a ``natural'' pcgs, the FamilyPcgs (see FamilyPcgs) with respect to which pcgs operations as described in chapter Polycyclic Groups are particularly efficient.

Let G be a polycyclic group with pcgs P = (g1, ¼, gn) and corresponding relative orders (r1, ¼, rn). Recall that the ri are positive integers or infinity and let I be the set of indices i with ri a positive integer. Then G has a finite presentation on the generators g1, ¼, gn with relations of the following form.

giri = gi+1a(i,i,i+1) ¼gna(i,i,n) for 1 £ i £ n and i Î I
gi-1 gj gi = gi+1a(i,j,i+1) ¼gna(i,j,n) for 1 £ i < j £ n

For infinite groups we need additionally

gi-1 gj-1 gi = gi+1b(i,j,i+1) ¼gnb(i,j,n) for 1 £ i < j £ n and j \not Î I
gi gj gi-1 = gi+1c(i,j,i+1) ¼gnc(i,j,n) for 1 £ i < j £ n and i \not Î I
gi gj-1 gi-1 = gi+1d(i,j,i+1) ¼gnd(i,j,n) for 1 £ i < j £ n and i, j, \not Î I

Here the right hand sides are assumed to be words in normal form; that is, for k Î I we have for all exponents 0 £ a(i,j,k), b(i,j,k), c(i,j,k), d(i,j,k) < rk.

A finite presentation of this type is called a power-conjugate presentation and a pc group is a polycyclic group defined by a power-conjugate presentation. Instead of conjugates we could just as well work with commutators and then the presentation would be called a power-commutator presentation. Both types of presentation are abbreviated as pc presentation. Note that a pc presentation is a rewriting system.

Clearly, whenever a group G with pcgs P is given, then we can write down the corresponding pc presentation. On the other hand, one may just write down a presentation on n abstract generators g1, ¼, gn with relations of the above form and define a group H by this. Then the subgroups Ci = ági, ¼, gn ñ of H form a subnormal series whose factors are cyclic or trivial. In the case that all factors are non-trivial, we say that the pc presentation of H is confluent. Note that GAP 4 can only work correctly with pc groups defined by a confluent pc presentation.

At the current level of implementation GAP can only deal with finite pc groups. This will be extended in near future.

Algorithms for pc groups use the methods for polycyclic groups described in chapter Polycyclic Groups.

Sections

  1. The family pcgs
  2. Elements of pc groups
  3. Pc groups versus fp groups
  4. Constructing Pc Groups
  5. Computing Pc Groups
  6. Saving a Pc Group
  7. Operations for Pc Groups
  8. 2-Cohomology and Extensions
  9. Coding a Pc Presentation
  10. Random Isomorphism Testing

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GAP 4 manual
February 2000