On Tue, May 17, 2016 at 9:04 AM, Dimitris Schinianakis <dsxoini...@gmail.com> wrote: > Dear all, > > I'm a beginner in group theory, currently trying to solve all the material > in the "A Book of abstract Algebra", by Pinter. > > I'd like to use Sage to verify some of the solutions but also to experiment > a little bit with Cayley Diagrams. >
I assume by Cayley diagram you mean Cayley graph, https://en.wikipedia.org/wiki/Cayley_graph > 1) Is there a way to draw a Cayley diagram in Sage by defining the group > and the equations for the generators? For example let > G={e,a,b,b^2,b^3,ab,ab^2,ab^3} and the generators satisfy a^2=e, b^4=e, > ba=ab^2. e is of course the identity element. > Not currently in Sage. However, in GAP, please check out the Grape and Digraphs packages at http://www.gap-system.org/Packages/packages.html The example below shows that your problem can be done in GAP using Grape. Please see the Grape manual for more options and examples. gap> f := FreeGroup( "a", "b" ); <free group on the generators [ a, b ]> gap> g := f / [ f.1^2, f.2^3, (f.1*f.2)^5 ]; <fp group on the generators [ a, b ]> gap> CayleyGraph(g); rec( adjacencies := [ [ 5, 7, 13 ] ], group := <permutation group of size 60 with 2 generators>, isGraph := true, isSimple := true, names := [ <identity ...>, b*a*(b*a*b)^2, a*b*a*(b*a*b)^2*a, b*a*(b*a*b)^2*b, b, b*(b*a)^2*b^2*a, b^2, (b*a)^2*b^2*a, b^2*a*(b*a*b)^2, a*(b*a*b)^2, a*(b*a*b)^2*b, (a*b)^2*b*a, a, a*b*a*(b*a*b)^2, b*a*(b*a*b)^2*a, (b*a*b)^2*a*b, b*(b*a)^2*b^2, b*a, a*(b*a*b)^2*a, a*b*(b*a)^2*b, (a*b)^2*b, b^2*a, (b*a)^2*b^2, (b*(b*a)^2)^2, a*b*a*(b*a*b)^2*b, a*b, a*b*(b*a)^2*b^2*a, a*(b*a*b)^2*a*b, a*b*(b*a)^2*b^2, a*b*a, (b*a*b)^2*a*b^2, b*(b*a)^2, b*a*b, b^2*a*b, (b*a*b^2*a)^2, (b*a)^2, a*b^2, (a*b)^3*b*a, (a*b^2*a*b)^2, (b*a*b)^2*a, b*(b*a)^2*b, b*a*b^2, ((a*b)^2*b)^2, a*b*(b*a)^2, (a*b)^2, b^2*a*b^2, (b*a)^2*b, (b*a*b^2*a)^2*b, (b*a*b)^2, (b*a*b)^2*b, b*a*b^2*a, a*b^2*a, (a*b)^3*b, (a*b^2*a*b)^2*a, a*b^2*a*b, ((a*b)^2*b)^2*a, (a*b)^2*a, (a*b^2)^2, (a*b)^3, ((a*b)^2*b)^2*a*b ], order := 60, representatives := [ 1 ], schreierVector := [ -1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2 ] ) > 2) I have a Cayley diagram with no labels, only connections and points are > depicted. I need to create the group table. How can I be sure that my table > is correct? Does a cayley diagram correspond always to only one group? > No, the Cayley graph does not correspond to only one group. In fact, one can select symmetric generators for ZZ/6ZZ (an abelian group) and S_3 (a non-abelian group) such that the associated Cayley graphs are isomorphic. By the way, while the GAP Forum is a place for questions and comments of general interest about GAP, the SageMath system has it's own support resources, including the SageMath support mailing list: http://www.sagemath.org/help-groups.html"; See also the related stackexchange post: http://math.stackexchange.com/questions/1789007/plot-cayley-graphs-for-generic-element-groups Hope this helps! > Regards > Dimitris > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
