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How do I find a presentation for classical matrix groups like PGL(2,q)?

asked 2013-01-25 17:23:53 -0500

oxeimon gravatar image

I know I can just take the group and do ".gens()", and that'll give me a list of generators, but is there a way to find the relations on them?

Also, is there a way to compute generators for kernels of group homomorphisms?

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answered 2013-01-27 18:53:49 -0500

benjaminfjones gravatar image

The usual way is to construct the canonical map from the appropriate free group to your group and then compute generators for the kernel. Free groups are supported by GAP and they have been the subject of some good recent work in Sage (see http://trac.sagemath.org/sage_trac/ti...) so this should be possible eventually (maybe in Sage v5.7?).

To compute the generators of a kernel you can do something like :

sage: G = PSL(2,7)
sage: D = G.direct_product(G)
sage: H = D[0]
sage: pr1 = D[3]
sage: pr1.kernel().gens()
[(11,15,13)(12,16,14), (9,10,14)(11,12,16)]

(from http://www.sagemath.org/doc/reference...)

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Asked: 2013-01-25 17:23:53 -0500

Seen: 298 times

Last updated: Jan 27 '13