[GAP Forum] Cayley diagrams for general groups

[David Joyner]

On Tue, May 17, 2016 at 9:04 AM, Dimitris Schinianakis
<dsxoiniana at 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
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