[GAP Forum] Generators of a group in ( n * n )Matrix fo
David Musyoka
davidmusyoka21 at yahoo.com
Thu Jul 9 16:01:57 BST 2020
Dear Dima, Dear Gap forum
Thank you very much,,have really learnt and continue to learn different approaches from the team.
Kind regards.
David
Sent from Yahoo Mail on Android
On Thu, 9 Jul 2020 at 17:41, dmitrii.pasechnik at cs.ox.ac.uk<dmitrii.pasechnik at cs.ox.ac.uk> wrote: Dear David,
for smallish groups you can directly compute the automorphism group
of a character table in GAP.
More pecisely, TableAutomorphisms.
gap> SmallGroupsInformation(81);
There are 15 groups of order 81.
They are sorted by their ranks.
1 is cyclic.
2 - 10 have rank 2.
11 - 14 have rank 3.
15 is elementary abelian.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.
gap> g81:=SmallGroup(81,15);
<pc group of size 81 with 4 generators>
gap> IsElementaryAbelian(g81); # an example of the group you are interested in
true
gap> t:=CharacterTable(g81);;
gap> a:=TableAutomorphisms(t,Irr(t),"closed");
<permutation group with 8 generators>
gap> OrbitLengths(a,[1..81]);
[ 1, 80 ]
(I think this illustrates the general picture - the trivial character
will be unique, in its own orbit, and the remaining characters are all
equivalent - which is not at all surprising, in view of
https://en.wikipedia.org/wiki/Pontryagin_duality)
Hope this helps,
Dima
On Sun, Jul 05, 2020 at 09:03:00AM +0000, David Musyoka wrote:
> Dear Dima and the GAP forum.
> Kindly appealing for your help in this;
> Given a group G and the group K=(q raised to the power of n) which is isomorphic to the Vector Space of Dimension n over GF(q),How do i compute the orbit lengths for the action of the group G on the set Irr(K) - the set of irreducible characters of K Using the GAP programme.
> Thanking you in advance.
> David.
> Sent from Yahoo Mail on Android
>
> On Mon, 29 Jun 2020 at 13:06, Dima Pasechnik<dmitrii.pasechnik at cs.ox.ac.uk> wrote: Dear all,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7 matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7 matrices using GAP prog.
>
>
> gap> G:=GeneralOrthogonalGroup(7,2); # use the fact that Sp(6,2)=O(7,2)
> GO(0,7,2)
> gap> o:=Orbits(G, GF(2)^7,OnLines);;
> gap> List(o,Size); # in this representation S8 fixes a hyperplane, not a vector
> [ 1, 63, 63, 1 ]
> gap> Gt:=Group(List(GeneratorsOfGroup(G),TransposedMat));
> <matrix group with 2 generators>
> gap> ot:=Orbits(Gt, GF(2)^7,OnLines);;
> gap> List(ot,Length); # the stabiliser of a vector in 3rd orbit is S8
> [ 1, 63, 36, 28 ]
> gap> s8:=Stabilizer(Gt,ot[3][1],OnLines); # here it is.
> <matrix group of size 40320 with 3 generators>
>
> Hope this helps
> Dima
>
> On Mon, Jun 29, 2020 at 07:32:51AM +0000, David Musyoka wrote:
> > Dear Forum, Dear Alexander Hulpke,
> > Given a group G, and and a vector space V of dimension n over GF(q), i am abe to compute the orbit Lengths of V under action of G using the following GAP commands:
> > V:=FullRowSpace(GF(q),n);Orb:=OrbitLengths(G,V)
> > My question is, how then do i compute the corresponding point stabilizers (which are subgroups of G) for the orbits using GAP.
> > Thank you team in advance.
> > David.
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 22:47, David Musyoka<davidmusyoka21 at yahoo.com> wrote: Dear Alexander Hulpke,Dear Forum.
> > Thank You very much.
> >
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 19:48, Hulpke,Alexander<Alexander.Hulpke at colostate.edu> wrote: Dear Forum, Dear Marc David Musyoka,
> >
> > On Jun 20, 2020, at 00:22, David Musyoka <davidmusyoka21 at yahoo.com> wrote:
> > Deat all,
> > Kind request to this team, i am new to GAP and i wish to be assisted in the following,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7 matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7 matrices using GAP prog.
> >
> > I wish that am assisted on how to execute the same step by step and the matrix generators for the two groups be listed.
> >
> >
> > Yes, I also often wish that someone would assist me in every step and provide me with the full result.
> > Anyhow, in this case (the algorithm is exponential time and attempts will fail if groups are too large, or if the subgroup to be embedded needs many generators) `IsomorphicSubgroups` seems to do the trick in a few minutes.
> > gap> G:=SP(6,2);Sp(6,2)gap> emb:=IsomorphicSubgroups(G,SymmetricGroup(8)); # finds embeddings from S8 into G[ [ (1,8)(2,7)(3,5)(4,6), (1,8,4,7,3)(5,6) ] -> [ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over GF2> ] ]gap> sub:=Image(emb[1]);<matrix group with 2 generators>
> > So there is one class of subgroups and `sub` is one (not necessarily the nicest one) representative.
> > Of course this is computational overkill. The more sensible way would be to produce the matrix representation (as reduced permutation representation), find the form that it stabilizes, and then conjugate that form to the one used for Sp.
> > Oh, here are the explicit matrix generators :-)
> > gap> GeneratorsOfGroup(sub);[ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over GF2> ]
> > (You can use `Print` or `Display` on each of them to see them with numbers.)
> > All the best,
> > Alexander Hulpke
> > -- Colorado State University, Department of Mathematics,
> > Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> > email: hulpke at colostate.edu,
> > http://www.math.colostate.edu/~hulpke
> >
> >
> >
> > _______________________________________________
> > Forum mailing list
> > Forum at gap-system.org
> > https://mail.gap-system.org/mailman/listinfo/forum
>
> Dear all,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7 matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7 matrices using GAP prog.
>
>
> gap> G:=GeneralOrthogonalGroup(7,2); # use the fact that Sp(6,2)=O(7,2)
> GO(0,7,2)
> gap> o:=Orbits(G, GF(2)^7,OnLines);;
> gap> List(o,Size); # in this representation S8 fixes a hyperplane, not a vector
> [ 1, 63, 63, 1 ]
> gap> Gt:=Group(List(GeneratorsOfGroup(G),TransposedMat));
> <matrix group with 2 generators>
> gap> ot:=Orbits(Gt, GF(2)^7,OnLines);;
> gap> List(ot,Length); # the stabiliser of a vector in 3rd orbit is S8
> [ 1, 63, 36, 28 ]
> gap> s8:=Stabilizer(Gt,ot[3][1],OnLines); # here it is.
> <matrix group of size 40320 with 3 generators>
>
> Hope this helps
> Dima
>
> On Mon, Jun 29, 2020 at 07:32:51AM +0000, David Musyoka wrote:
> > Dear Forum, Dear?Alexander Hulpke,
> > Given a group G, and and a vector space V of dimension n over GF(q), i am abe to compute the orbit Lengths of V under action of G using the following GAP commands:
> > V:=FullRowSpace(GF(q),n);Orb:=OrbitLengths(G,V)
> > My question is, how then do i compute the corresponding point stabilizers (which?are subgroups of G)?for the orbits using GAP.
> > Thank you team in advance.
> > David.
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 22:47, David Musyoka<davidmusyoka21 at yahoo.com> wrote: Dear?Alexander Hulpke,Dear Forum.
> > Thank You very much.
> >
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 19:48, Hulpke,Alexander<Alexander.Hulpke at colostate.edu> wrote: Dear Forum, Dear Marc David Musyoka,
> >
> > On Jun 20, 2020, at 00:22, David Musyoka <davidmusyoka21 at yahoo.com> wrote:
> > Deat all,
> > Kind request to this team, i am new to GAP and i wish to be assisted in the following,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7 matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7 matrices using GAP prog.
> >
> > I wish that am assisted on how to execute the same step by step and the matrix generators for the two groups be listed.
> >
> >
> > Yes, I also often wish that someone would assist me in every step and provide me with the full result.
> > Anyhow, in this case (the algorithm is exponential time and attempts will fail if groups are too large, or if the subgroup to be embedded needs many generators) `IsomorphicSubgroups` seems to do the trick in a few minutes.
> > gap> G:=SP(6,2);Sp(6,2)gap> emb:=IsomorphicSubgroups(G,SymmetricGroup(8)); # finds embeddings from S8 into G[ [ (1,8)(2,7)(3,5)(4,6), (1,8,4,7,3)(5,6) ] ->? ? [ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over GF2> ] ]gap> sub:=Image(emb[1]);<matrix group with 2 generators>
> > So there is one class of subgroups and `sub` is one (not necessarily the nicest one) representative.
> > Of course this is computational overkill. The more sensible way would be to produce the matrix representation (as reduced permutation representation), find the form that it stabilizes, and then conjugate that form to the one used for Sp.
> > Oh, here are the explicit matrix generators :-)
> > gap> GeneratorsOfGroup(sub);[ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over GF2> ]
> > (You can use `Print` or `Display` on each of them to see them with numbers.)
> > All the best,
> > ? Alexander Hulpke
> > -- Colorado State University,?Department of Mathematics,
> > Weber Building, 1874 Campus?Delivery, Fort Collins, CO 80523-1874, USA
> > email:?hulpke at colostate.edu,?
> > http://www.math.colostate.edu/~hulpke
> >
> >
> >
> > _______________________________________________
> > Forum mailing list
> > Forum at gap-system.org
> > https://mail.gap-system.org/mailman/listinfo/forum
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