ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 27 Jan 2013 18:53:49 -0600How do I find a presentation for classical matrix groups like PGL(2,q)?http://ask.sagemath.org/question/9743/how-do-i-find-a-presentation-for-classical-matrix-groups-like-pgl2q/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?Fri, 25 Jan 2013 17:23:53 -0600http://ask.sagemath.org/question/9743/how-do-i-find-a-presentation-for-classical-matrix-groups-like-pgl2q/Answer by benjaminfjones for <p>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?</p>
<p>Also, is there a way to compute generators for kernels of group homomorphisms?</p>
http://ask.sagemath.org/question/9743/how-do-i-find-a-presentation-for-classical-matrix-groups-like-pgl2q/?answer=14495#post-id-14495The 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/ticket/12339) 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/sage/groups/perm_gps/permgroup_morphism.html)
Sun, 27 Jan 2013 18:53:49 -0600http://ask.sagemath.org/question/9743/how-do-i-find-a-presentation-for-classical-matrix-groups-like-pgl2q/?answer=14495#post-id-14495