| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/polenta/doc/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 10.3.2025 mit Größe 3 kB |
|
Quelle example.xml Sprache: XML |
|||||||||
|
<Chapter Label="An example application">
<Heading>An example application</Heading> In this section we outline three example computations with functions from the previous chapter. <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <Section Label="Presentation for rational matrix groups"> <Heading>Presentation for rational matrix groups</Heading> <Example><![CDATA[ gap> mats := [ [ [ 1, 0, -1/2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 1/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -1/2, -3, 7/6 ], [ 0, 1, -1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ], [ [ -1, 3, 3, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ] ]; gap> G := Group( mats ); <matrix group with 5 generators> # calculate an isomorphism from G to a pcp-group gap> nat := IsomorphismPcpGroup( G );; gap> H := Image( nat ); Pcp-group with orders [ 2, 2, 3, 5, 5, 5, 0, 0, 0 ] gap> h := GeneratorsOfGroup( H ); [ g1, g2, g3, g4, g5, g6, g7, g8, g9] gap> mats2 := List( h, x -> PreImage( nat, x ) );; # take a random element of G gap> exp := [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ];; gap> g := MappedVector( exp, mats2 ); [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] # map g into the image of nat gap> i := ImageElm( nat, g ); g1*g2*g3*g4*g9 # exponent vector gap> Exponents( i ); [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ] # compare the preimage with g gap> PreImagesRepresentative( nat, i ); [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] gap> last = g; true ]]></Example> </Section> <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <Section Label="Modules series"> <Heading>Modules series</Heading> <Example><![CDATA[ gap> gens := [ [ [ 1746/1405, 524/7025, 418/1405, -77/2810 ], [ 815/843, 899/843, -1675/843, 415/281 ], [ -3358/4215, -3512/21075, 4631/4215, -629/1405 ], [ 258/1405, 792/7025, 1404/1405, 832/1405 ] ], [ [ -2389/2810, 3664/21075, 8942/4215, -35851/16860 ], [ 395/281, 2498/2529, -5105/5058, 3260/2529 ], [ 3539/2810, -13832/63225, -12001/12645, 87053/50580 ], [ 5359/1405, -3128/21075, -13984/4215, 40561/8430 ] ] ]; gap> H := Group( gens ); <matrix group with 2 generators> gap> RadicalSeriesSolvableMatGroup( H ); [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 79/138 ], [ 0, 1, 0, -275/828 ], [ 0, 0, 1, -197/414 ] ], [ [ 1, 0, -3, 2 ], [ 0, 1, 55/4, -55/8 ] ], [ [ 1, 4/15, 2/3, 1/6 ] ], [ ] ] ]]></Example> </Section> <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <Section Label="Triangularizable subgroups"> <Heading>Triangularizable subgroups</Heading> <Example><![CDATA[ gap> G := PolExamples(3); <matrix group with 2 generators> gap> GeneratorsOfGroup( G ); [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] gap> subgroups := SubgroupsUnipotentByAbelianByFinite( G ); rec( T := <matrix group with 2 generators>, U := <matrix group with 4 generators> ) gap> GeneratorsOfGroup( subgroups.T ); [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] # so G is triangularizable! ]]></Example> </Section> </Chapter> ¤ Dauer der Verarbeitung: 0.12 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
|
2026-03-28