[GAP Forum] Generators of a group in ( n * n )Matrix form

[Hulpke,Alexander]

Dear Forum, Dear Marc David Musyoka,

On Jun 20, 2020, at 00:22, David Musyoka <davidmusyoka21 at yahoo.com<mailto: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<mailto:hulpke at colostate.edu>,
http://www.math.colostate.edu/~hulpke