43.4 Constructing Pc Groups

creates a PcGroup P from an FpGroup (see Chapter Finitely Presented Groups) G whose presentation is polycyclic. The resulting group P has generators corresponding to the generators of G. They are printed in the same way as generators of G, but they lie in a different family. If the pc presentation of G is not confluent, an error message occurs.

gap> F := FreeGroup("a","b","c","d");;
gap> a := F.1;; b := F.2;; c := F.3;; d := F.4;;         
gap> rels := [a^2, b^3, c^2, d^2, Comm(b,a)/b, Comm(c,a)/d, Comm(d,a),
>             Comm(c,b)/(c*d), Comm(d,b)/c, Comm(d,c)];
[ a^2, b^3, c^2, d^2, b^-1*a^-1*b*a*b^-1, c^-1*a^-1*c*a*d^-1, d^-1*a^-1*d*a, 
  c^-1*b^-1*c*b*d^-1*c^-1, d^-1*b^-1*d*b*c^-1, d^-1*c^-1*d*c ]
gap> G := F / rels; 
<fp group on the generators [a,b,c,d]>
gap> H := PcGroupFpGroup( G );
<pc group of size 24 with 4 generators>

Equivalently to the above method one can initiate a collector of a pc group by hand and use it to define a pc group. In GAP there are different collectors for different collecting strategies; at the moment, there are two collectors to choose from: the single collector for finite pc groups and the combinatorial collector for finite p-groups. See Sims94 for further information on collecting strategies.

A collector is initiated by underlying free group to the pc presented group and the relative orders of the pc series. Then one adds the right hand sides of the power and the commutator or conjugate relations one by one. Note that omitted relators are assumed to be trivial.

Then the right hand sides of the pc presentation have to be declared. Let f₁, ¼, f_n be the generators of the underlying free group fgrp.

set the conjugate f_j^f_i to equal w where w is a word in f_i+1, ¼, f_n and i < j.

set the commutator of f_j and f_i to equal w where w is a word in f_i+1, ¼, f_n and i < j.

set the power f_i^r_i to equal w where w is a word in f_i+1, ¼, f_n.

Finally, the collector has to be converted to a group.

creates a group from a rewriting system. In the first version it is checked whether the rewriting system is confluent, in the second version this is assumed to be true.

checks whether the pc group G has been build from a collector with a confluent power-commutator presentation.

gap> F := FreeGroup( 2 );;
gap> coll1 := SingleCollector( F, [2,3] );
<<single collector, 8 Bits>>
gap> SetConjugate( coll1, 2, 1, F.2 );  
gap> SetPower( coll1, 1, F.2 );
gap> G1 := GroupByRws( coll1 );
<pc group of size 6 with 2 generators>
gap> IsConfluent(G1);
true
gap> IsAbelian(G1);
true

gap> coll2 := SingleCollector( F, [2,3] );
<<single collector, 8 Bits>>
gap> SetConjugate( coll2, 2, 1, F.2^2 );
gap> G2 := GroupByRws( coll2 );
<pc group of size 6 with 2 generators>
gap> IsAbelian(G2); 
false

With the above methods a pc group with arbitrary defining pcgs can be constructed. However, for almost all applications within GAP we need to have a pc group whose defining pcgs is a prime order pcgs. Hence the following functions are useful.

returns an isomorphism from G onto an isomorphic PC group whose family pcgs is a prime order pcgs.

returns the range of IsomorphismRefinedPcGroup(G).

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