Dear Alexander Konovalov, Thank you very much for your description with some execution examples!
I tried to run Copy and paste the example of you.It's very interesting. I have not read the document you taught me yet, but I understand the point. I think that is similar to polymorphism of object-oriented programming languages. I was easier to read the document and source codes thanks to your explanation. Thank you very much. With best regards buynnnmmm1 ----- Original Message ----- > From: Alexander Konovalov <al...@mcs.st-andrews.ac.uk> > To: buynnnm...@yahoo.co.jp > Cc: GAP Forum <fo...@gap-system.org> > Date: 2014/9/18, Thu 06:19 > Subject: Re: [GAP Forum] Is it possible to step through the program, like GNU > GDB debugger, against built-in functions(ex. DerivedSubgroup, > ClosureSubgroupNC )? > > > On 17 Sep 2014, at 14:51, buynnnm...@yahoo.co.jp wrote: > >> Dear Alexander Konovalov, >> >> Thank you very much for your help. >> >>> Not really - the method for IsSolvable did not evolve from the code > similar to >>> myIsSolvable at all. >>> >>> Perhaps the key is to read about GAP method selection and learn the > concept of >>> methods as bundles of functions: >>> >>> http://www.gap-system.org/Manuals/doc/tut/chap8.html#X7AEED9AB824CD4DA >>> >>> - that is, IsSolvable(G) will select the best available method to apply > to G, >>> taking into account what's known about G at the moment. With > myIsSolvable, >>> you enforce the calculation of DerivedSeries, while some of the methods > may need >>> not to know the derived series at all to give an answer. The profile > below just >>> illustrates this, since the number of methods involved in the calls to > GAP's >>> IsSolvable is much smaller. >> >> IsSolvable in lib/grp.gi is the same as the method for determining whether > or not solvable group I have learned . >> I'll try to understand that the method for determining whether or not > solvable group I have learned is the same the built-in IsSolvable function. >> I think it will be the good study of group theory . >> I'll try to be able to understand "Operations and Methods of > GAP" with the document, too. >> Thank you very much for your help. >> >> With best regards >> buynnnmmm1 > > Note also that several lines above there is an immediate method which > expresses the famous Feit–Thompson theorem stating that any group of > odd order is solvable: > > > InstallImmediateMethod( IsSolvableGroup, IsGroup and HasSize, 10, > function( G ) > G:= Size( G ); > if IsInt( G ) and G mod 2 = 1 then > return true; > fi; > TryNextMethod(); > end ); > > > The method to which you refer is generic, as its description says - like a > fallback method that is deemed to work for any group: > > > InstallMethod( IsSolvableGroup, > "generic method for groups", > [ IsGroup ], > function ( G ) > local S; # derived series of <G> > > # compute the derived series of <G> > S := DerivedSeriesOfGroup( G ); > > # the group is solvable if the derived series reaches the trivial group > return IsTrivial( S[ Length( S ) ] ); > end ); > > > You may trace how the method selection works adding the optional argument > "full". > For example, for S(3) the method ``IsSolvableGroup: for permgrp'' will > be used > quite early: > > gap> ApplicableMethod(IsSolvableGroup,[SymmetricGroup(3)],"full"); > #I Searching Method for IsSolvableGroup with 1 arguments: > #I Total: 11 entries > #I Method 1: ``IsSolvableGroup: system getter'', value: 2*SUM_FLAGS+21 > #I - 1st argument needs [ "Tester(IsSolvableGroup)" ] > #I Method 2: ``IsSolvableGroup: handled by nice monomorphism: > Attribute'', value: 360 > #I - 1st argument needs [ "IsHandledByNiceMonomorphism", > "Tester(IsHandledByNiceMonomorphism)" ] > #I Method 3: ``IsSolvableGroup: for permgrp'', value: 48 > #I Function Body: > function ( G ) > local pcgs; > pcgs := TryPcgsPermGroup( G, false, false, true ); > if IsPcgs( pcgs ) then > SetIndicesEANormalSteps( pcgs, pcgs!.permpcgsNormalSteps ); > SetIsPcgsElementaryAbelianSeries( pcgs, true ); > if not HasPcgs( G ) then > SetPcgs( G, pcgs ); > fi; > if not HasPcgsElementaryAbelianSeries( G ) then > SetPcgsElementaryAbelianSeries( G, pcgs ); > fi; > return true; > else > return false; > fi; > return; > endfunction( G ) ... end > gap> > > To give an example of a group for which method selection descends to the > fallback method, let's construct a dihedral group of order 16 as a finitely > presented group: > > gap> f:=FreeGroup("x","y"); > <free group on the generators [ x, y ]> > gap> r:=ParseRelators(GeneratorsOfGroup(f),"x^8=y^2=1,yxy=x^-1"); > [ x^8, y^2, (x^-1*y^-1)^2 ] > gap> G:=f/r; > <fp group on the generators [ x, y ]> > > Now after some information about other available methods, > the last one is precisely the generic method: > > gap> ApplicableMethod(IsSolvableGroup,[G],"full"); > #I Searching Method for IsSolvableGroup with 1 arguments: > #I Total: 11 entries > #I Method 1: ``IsSolvableGroup: system getter'', value: 2*SUM_FLAGS+21 > #I - 1st argument needs [ "Tester(IsSolvableGroup)" ] > #I Method 2: ``IsSolvableGroup: handled by nice monomorphism: > Attribute'', value: 360 > #I - 1st argument needs [ "IsHandledByNiceMonomorphism", > "Tester(IsHandledByNiceMonomorphism)" ] > #I Method 3: ``IsSolvableGroup: for permgrp'', value: 48 > #I - 1st argument needs [ "CategoryCollections(IsPerm)" ] > #I Method 4: ``IsSolvableGroup: for AffineCrystGroup, via PointGroup'', > value: 43 > #I - 1st argument needs [ "IsAffineCrystGroupOnLeftOrRight", > "Tester(IsAffineCrystGroupOnLeftOrRight)" ] > #I Method 5: ``IsSolvableGroup: for rational matrix groups (Polenta)'', > value: 40 > #I - 1st argument needs [ "IsRationalMatrixGroup", > "Tester(IsRationalMatrixGroup)" ] > #I Method > 6: ``IsSolvableGroup: for matrix groups over a finte field (Polenta)'', > value: 38 > #I - 1st argument needs > [ > > "CategoryCollections(CategoryCollections(CategoryCollections(IsNearAdditiveElementWithIn\ > verse)))", > > "CategoryCollections(CategoryCollections(CategoryCollections(IsAdditiveElement)))", > > > > "CategoryCollections(CategoryCollections(CategoryCollections(IsMultiplicativeElement)))\ > ", > "CategoryCollections(CategoryCollections(CategoryCollections(IsFFE)))" > ] > #I Method 7: ``IsSolvableGroup: fallback method to test conditions'', > value: 38 > #I - 1st argument needs > [ > > "CategoryCollections(CategoryCollections(CategoryCollections(IsNearAdditiveElementWithIn\ > verse)))", > > "CategoryCollections(CategoryCollections(CategoryCollections(IsAdditiveElement)))", > > > > "CategoryCollections(CategoryCollections(CategoryCollections(IsMultiplicativeElement)))\ > ", > "CategoryCollections(CategoryCollections(CategoryCollections(IsCyclotomic)))" > ] > #I Method 8: ``IsSolvableGroup'', value: 27 > #I - 1st argument needs [ "Tester(Size)" ] > #I Method 9: ``IsSolvableGroup: for direct products'', value: 26 > #I - 1st argument needs [ "Tester(DirectProductInfo)" ] > #I Method 10: ``IsSolvableGroup: generic method for groups'', value: 25 > #I Function Body: > function ( G ) > local S; > S := DerivedSeriesOfGroup( G ); > return IsTrivial( S[Length( S )] ); > endfunction( G ) ... end > > However, if you create D_16 as permutation group, ApplicableMethod > will return the same ``IsSolvableGroup: for permgrp'' as for S(3) > above: > > gap> G:=Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]); > Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]) > gap> ApplicableMethod(IsSolvableGroup,[G],"full"); > ... > #I Method 3: ``IsSolvableGroup: for permgrp'', value: 48 > #I Function Body: > function ( G ) > local pcgs; > pcgs := TryPcgsPermGroup( G, false, false, true ); > > ... > > so as you may see, different methods may be used dependently on how the > group is represented. Hope that clarifies some more details! > > Alexander > _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
