Clearly, the generators of a power-conjugate presentation of a pc group G form a pcgs of the pc group. This pcgs is called the family pcgs.
FamilyPcgs( grp ) A
IsFamilyPcgs( pcgs ) P
InducedPcgsWrtFamilyPcgs( grp ) A
IsParentPcgsFamilyPcgs( pcgs ) P
This property indicates that the pcgs pcgs is induced with respect to a family pcgs.
In GAP 3 the family pcgs had been the only pcgs allowed for a pc group. Note that this has changed in GAP 4 where a pc group may have several independent polycyclic generating sequences.
However, the elementary operations for a non-family pcgs may not be as efficient as the elementary operations for the family pcgs.
This can have a notable influence on the performance of algorithms for
polycyclic groups. Many algorithms require a pcgs that corresponds to an
elementary abelian series (see PcgsElementaryAbelianSeries) or even a
special pcgs (see Special Pcgs). If the family pcgs has the required
properties, it will be used for these purposes, if not GAP has to work
with respect to a new pcgs which is not the family pcgs and thus takes
longer for elementary calculations like ExponentsOfPcElement.
Therefore, if the family pcgs chosen for arithmetic is not of importance it
might be worth to change to another, nicer, pcgs to speed up calculations.
This can be achieved, for example, by using the Range of the isomorphism
obtained by IsomorphismSpecialPcGroup (see IsomorphismSpecialPcGroup).
GAP 4 manual