Polycyclic Groups [HB 22]

 

Changes:

• A new symbolic collector with a much better complexity in the exponents occurring in element vectors has been installed. The speed up can reach several orders of magnitude in typical examples and virtually all computations with polycyclic groups benefit from this improvement.

• The limitation that exponents in a polycyclic presentation and exponents in element vectors be restricted to values less than 2³⁰ has been removed.

• A function PresentationIsSmall has been added, which enables the user to check whether big integer arithmetic actually is required for a polycyclic presentation. This may be relevant for some category transfers.

• The constructor PolycyclicGroup now returns a group in the category GrpPC or a group in the category GrpGPC, depending on the presentation passed and on the values of a parameter. This simplifies the construction of polycyclic groups and provides a common interface for the closely related group categories GrpPC and GrpGPC.

• The category transfer functions FPGroup and PCGroup now check for small presentations, since big integers are not supported by the target categories.

• Homomorphisms with a domain of type GrpGPC are now checked to be well-defined. (This feature can be turned off.)

• The (trivial) kernels of homomorphisms returned by category transfer functions are now properly embedded in the domain.

• The functions CosetKernel and CosetAction (if the kernel of the coset action is actually assigned to a variable) now in all cases successfully compute the kernel of the coset action.

• The kernels of homomorphisms with a domain of type GrpGPC are no longer computed during the construction of the map, but are computed only when the kernel is actually accessed. This speeds up definitions of homomorphisms and several functions returning or internally using homomorphisms.

New features for general polycyclic groups:

• For homomorphisms with a domain of type GrpGPC, an attempt is made to compute the kernel when the kernel is accessed directly or indirectly as a consequence of another function call. Computing kernels currently is possible for codomains of the following data types: GrpGPC, GrpPC, GrpPerm and GrpAb, as well as for codomains of the type GrpMat, provided that the image is finite.

  As a consequence, computing preimages of subgroups of the codomain is now possible in these situations.

• The new function FreeNilpotentGroup returns the free nilpotent group of given rank and class as a polycyclic group.
• FreeAbelianGroup returns the free abelian group of given rank as a polycyclic group.
• The new function ElementaryAbelianGroup returns the elementary abelian group of order pⁿ for a given prime p and a given integer n as a polycyclic group.
• The functions AbelianQuotient, AbelianQuotientInvariants, ElementaryAbelianQuotient and FreeAbelianQuotient provide information about the structure of abelian quotients of a polycyclic group.
• The new functions FittingSubgroup, FittingSeries and FittingLength compute the Fitting subgroup, the Fitting series and the length of the Fitting series of a polycyclic group.
• The new function UpperCentralSeries computes the upper central series of a polycyclic group.
• The function Centre computes the centre of a polycyclic group.
• EFASeries computes a normal series for a polycyclic group, the factors of which are either elementary abelian p-groups or free abelian groups (elementary or free abelian series; efa-series).
• The new function SemisimpleEFASeries computes a normal series for a polycyclic group G, the factors of which are either elementary abelian p-groups which are semisimple as ${\bf F}_p[G]$-modules or free abelian groups which are semisimple as ${\bf Q}[G]$-modules (semisimple efa-series).
• The new functions EFAModules and SemisimpleEFAModules return the sequence of ${\bf F}_p[G]$- or ${\bf Z}[G]$-modules determined by the action of G on the factors of an efa-series of G or on the factors of a semisimple efa-series of G, respectively.
• EFAModuleMaps and SemisimpleEFAModuleMaps return a sequence of maps from the (non-trivial) subgroups of an efa-series of G or of a semisimple efa-series of G, respectively, onto the ${\bf F}_p[G]$- or ${\bf Z}[G]$-modules determined by the action of G on the factors of the efa-series or the semisimple efa-series, respectively.
• Four new functions GModule compute the R[G]-module induced by the action of a polycyclic group G on the maximal abelian quotient ( $R = {\bf Z}$) or the maximal p-elementary abelian quotient ( $R = {\bf F}_p$), respectively, of a normal subgroup or of a section of G. The new function GModulePrimes computes the set of primes for which the R[G]-module induced as described above is non-trivial and the dimensions of the corresponding R[G]-modules.

New features for nilpotent polycyclic groups:

• Two new functions Centraliser compute the centraliser of an element and of a subgroup of a nilpotent polycyclic group.
• The new function Normaliser computes the normaliser of a polycyclic group in another polycyclic group. This function requires the existence of a nilpotent covering group.
• The new function IsSelfNormalising tests whether a subgroup of a nilpotent polycyclic group is self-normalising in the supergroup.
• Two new functions IsConjugate test whether two elements or two subgroups, respectively, of a nilpotent covering group are conjugate under the action of a subgroup. If so, a conjugating element is computed.

Bug Fixes:

• Some bugs and memory leaks have been fixed. They involve intersection of subgroups, consistency check for presentations, natural epimorphism for quotients, IsCyclic and AbelianGroup.