labelsection_present
Groups defined by polycyclic presentations are called PcpGroups in GAP. We refer to the Polycyclic manual polycyclic for further background.
Suppose that a collection of matrices of GL(d,R) is given, where the ring R is either Q,Z or a finite field. Let G be the group which is generated by these matrices. If the group G is polycyclic, then the following functions determine a PcpGroup isomorphic to G.
PcpGroupByMatGroup(
G )
IsomorphismPcpGroup(
G )
G is a subgroup of GL(d,R) where R=Q,Z or Fq. If G is polycyclic, then these functions determine a PcpGroup isomorphic to G and an isomorphism onto this group. If G is not polycyclic, then there are two cases: If R=Z or Fq, then the algorithm returns 'fail'. In case that R=Q the algorithm may return 'fail' or may not terminate.
Note, that the method tt IsomorphismPcpGroup, installed in this package, cannot be applied directly to a group given by the function ttAlmostCrystallographicGroup. Please use ttPOL_AlmostCrystallographicGroup (with the same parameters as ttAlmostCrystallographicGroup) instead.
Image(
map )
ImageElm(
map,
elm )
ImagesSet(
map,
elms )
PreImagesRepresentative(
map,
pcpelm )
Here map is an isomorphism from a polycyclic matrix group G onto a PcpGroup H calculated by ttIsomorphismPcpGroup(G). These functions can be used to compute with such an isomorphism. If the input elm is an element of G, then the function tt ImageElm can be used to compute the image of elm under map. If elm is not contained in G then the function ttImageElm returns 'fail'. The input pcpelm is an element of H.
IsSolvableGroup(
G )
IsSolvableMatGroup(
G )
G is a subgroup of GL(d,R) where R=Q,Z or Fq. This function tests if G is solvable and returns 'true' or 'false'.
IsPolycyclicMatGroup(
G )
G is a subgroup of GL(d,R) where R=Q,Z or Fq. This function tries to test if G is polycyclic. If G is polycyclic, then it returns 'true'. If G is not polycyclic, then there are two cases: If R=Z or Fq, then the function returns 'false'. In case that R=Q the algorithm may return 'false' or may not terminate.
Let G be a finitely generated solvable subgroup of GL(d,Q). The vectors
space Qd is a module for the algebra Q[G]. The following
functions provide the possiblity to compute certain module series of
Qd. Recall that the radical RadG(Qd) is definied to be the
intersection of maximal Q[G]-submodules of Qd. Further the
radical series
|
RadicalSeriesSolvableMatGroup(
G )
return the radical series for the solvable rational matrix group G.
A module is said to homogeneous if it is the direct sum of irreducible and isomorphic submodules. A radical series of Qd can be refinied to a homogeneous series. That is a submodule series such that the factors are homogeneous.
HomogeneousSeriesAbelianMatGroup(
G )
returns the homogeneous series for the abelian rational matrix group G.
Further a homogeneous series can refined to a composition series. That is a submodule series such that the factors are irreducible.
CompositionSeriesAbelianMatGroup(
G )
returns the composition series for the abelian rational matrix group G.
PolExamples(
l )
Returns some examples for polycyclic rational matrix groups, where l is an integer between 1 and 24. These can be used to test the functions in this package.
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