This manual describes the matgrp package. This package for GAP provides an interface for Solvable Radical (or Fitting-free) algorithms for matrix groups, using a recognition tree. The underlying rings supported are finite fields and residue class rings of the integers modulo a (possibly composite) number.
The package also implements routines that make this data structure work, notably routines for the epimorphism on the radical factor, data structures for subgroups, as well as for polyclic generating sets for solvable matrix (sub)groups.
The code is still experimental, but should be as stable as the underlying packages.
It is copyright (C) by Alexander Hulpke and released under version 2 and version 3 of the GPL. (It is a bad habit to include copies of the GPL in software, as it can easily be found under http://gnu.org/licenses/gpl.html.)
If you were able to install the prerequisite package recog, you will have no problems in installing matgrp, which is plain GAP code.
There is basically just one operation of interest. The package implements a
method for FittingFreeLiftSetup that applies to matrix groups over finite
fields, or over residue class rings of the integers.
gap> gens:=[ [ [ -28, 3, 9 ], [ 0, -1, -6 ], [ 3, 0, 1 ] ], > [ [ -1, 0, 0 ], [ -9, 1, 3 ], [ -3, 0, -1 ] ], > [ [ 0, 0, 1 ], [ 1, 0, 9 ], [ 0, 1, 0 ] ] ]; gap> g:=Group(List(gens,x->x*One(Integers mod 89))); <matrix group with 3 generators> gap> ffs:=FittingFreeLiftSetup(g);; gap> Size(Image(ffs.factorhom)); 704880 gap> RelativeOrders(ffs.pcgs); [ 11, 2, 2, 2, 89, 89 ] gap> h:=Group(List(gens,x->x*One(Integers mod 2403))); <matrix group with 3 generators> gap> ffs:=FittingFreeLiftSetup(h);; gap> Size(Image(ffs.factorhom)); 704880 gap> RelativeOrders(ffs.pcgs); [ 3, 2, 2, 11, 2, 2, 2, 89, 89, 3, 3, 3, 3, 3, 3, 3, 3, 3 ]
Once this data structure has been computed, library functionality that uses
it (that is newer versions of Solvable Radical/Fitting free algorithms)
become available. (Note that method selection does not yet catch all these
cases and that in particular FittingFreeLiftSetup needs to be called
explicitly at the start to avoid falling back on generic library methods.)
gap> cl:=ConjugacyClasses(g);; gap> Collected(List(cl,Size)); [ [ 1, 1 ], [ 7920, 1 ], [ 7921, 87 ], [ 704880, 1 ], [ 712890, 87 ], [ 62029440, 1 ], [ 62037272, 3916 ], [ 62734320, 174 ], [ 63447210, 3741 ] ] gap> RepresentativeAction(g,g.1,g.1^g.2); [ [ Z(89)^21, Z(89)^60, 0*Z(89) ], [ Z(89)^22, Z(89)^54, Z(89)^80 ], [ Z(89)^86, Z(89)^34, Z(89)^35 ] ] gap> g.1^last=g.1^g.2; true gap> m:=MaximalSubgroupClassReps(h); [ <matrix group of size 38683802512903680 with 20 generators>, <matrix group of size 10550127958064640 with 20 generators>, <matrix group of size 29012851884677760 with 19 generators>, <matrix group of size 14651105610240 with 19 generators>, <matrix group of size 4298200279211520 with 18 generators>, <matrix group of size 58025703769355520 with 20 generators>, <matrix group of size 1289460083763456 with 20 generators>, <matrix group of size 29635190893440 with 20 generators>, <matrix group of size 28976631095808 with 20 generators> ] gap> s:=HallViaRadical(g,[2])[1]; <matrix group of size 128 with 7 generators> gap> n:=NormalizerViaRadical(g,s); <matrix group of size 1408 with 7 generators>
To improve performance, the package provides a rudinemntary data type for matrices over residue class rings (which in the long run ought to be replaced by a better implementation in the main system), that yields a somewhat faster multiplication than using lists of lists.
The command ZmodnzMat(ring,intmat) will create a matrix in this compact
format, for best performance convert matrices into this format before
creating a group.
gap> r:=Integers mod 2403; (Integers mod 2403) gap> gens:=List(gens,x->ZmodnZMat(r,x)); [ [ [ 2375, 3, 9 ], [ 0, 2402, 2397 ], [ 3, 0, 1 ] ]*ZmodnZObj(1,2403), [ [ 2402, 0, 0 ], [ 2394, 1, 3 ], [ 2400, 0, 2402 ] ]*ZmodnZObj(1,2403), [ [ 0, 0, 1 ], [ 1, 0, 9 ], [ 0, 1, 0 ] ]*ZmodnZObj(1,2403) ] gap> h:=Group(gens); <matrix group with 3 generators> gap> ffs:=FittingFreeLiftSetup(h);;
The ultimate goal of this package surely must be to have GAP work with matrix groups in the same way as with permutation groups, simply using recognition as underlying structure. This is not yet feasible at the moment and will require some larger changes to the system:
recog package does not provide a verification routine that
would guarantee a correct recognition tree.
In certain situations the current version of the code may produce (one-off) error messages. Some reasons for this are:
In general, if a calculation in a matgrp routine produces an error, it can
be worth trying the command again, more often than not it will run fine. If
errors arise repeatedly, please inform the package author.
Some error messages issued by the package are:
Error, Wrong size -- set doall optiondoall option to true to enforce use of a proven
generating set.
no vector found
some discrepancy happened
layerlen
matgrp manual