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Cosets Generated by Product of Generators

asked 2013-10-22 15:42:04 -0600

anonymous user

Anonymous

updated 2013-10-24 12:55:12 -0600

I have created the following group:

sage: F.<x,y> = FreeGroup()
sage: G = F / [x^3, y^2, (x*y)^5]
sage: G
Finitely presented group < x, y | x^3, y^2, (x*y)^5 >

I would like to list the distinct cosets generated by <x*y>. I think there should be 12, but I have tried different code with no luck.

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answered 2013-10-24 12:43:21 -0600

updated 2013-10-24 16:15:31 -0600

Try

sage: F.<x,y> = FreeGroup()
sage: G = F / [x^2, y^3, (x*y)^5]
sage: xx = G(x)
sage: yy = G(y)
sage: (xx*yy)^5 == G.one()
True
sage: (xx*yy) == G.one()
False

Now knowing that (x*y) is not the identity in the quotient and that its order divides 5 you have no choice.

note: your group G is isomorphic to A5 by sending x to (1,2)(3,4) and y to (1,3,5).

edit: To actually get the coset one can use the functionalities of GAP

sage: U = G.subgroup([G(x*y)])
sage: for rc in G.gap().RightCosets(U.gap()): print rc
RightCoset(Group( [ x*y ] ),<identity ...>)
RightCoset(Group( [ x*y ] ),x)
RightCoset(Group( [ x*y ] ),y)
RightCoset(Group( [ x*y ] ),y*x)
RightCoset(Group( [ x*y ] ),y*x*y)
RightCoset(Group( [ x*y ] ),(y*x)^2)
RightCoset(Group( [ x*y ] ),y*x*y^2)
RightCoset(Group( [ x*y ] ),(y*x)^2*y)
RightCoset(Group( [ x*y ] ),(y*x)^2*y^2)
RightCoset(Group( [ x*y ] ),(y*x)^2*y^2*x)
RightCoset(Group( [ x*y ] ),y*x*(y*x*y)^2)
RightCoset(Group( [ x*y ] ),y*x*(y*x*y)^2*x)
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Comments

Thank you for taking a look at this...I do see that the group is isomorphic to A5, but I was hoping that sage could actually give me a list of representatives from the 12 cosets generated by xy. What I mean is...is there a way to give me the list: x<xy>, y<xy>, yx^2y<xy>, and so on?

LearningSAGE gravatar imageLearningSAGE ( 2013-10-24 15:07:23 -0600 )edit

It makes sense for me to show you what code I thought would work maybe you can tell me what I am doing wrong. After implementing the code you gave me above, I used the following: xy=G(x)*G(y) P = G.subgroup([xy]) G.cosets(P, side='left')

LearningSAGE gravatar imageLearningSAGE ( 2013-10-24 15:40:09 -0600 )edit

The method "cosets" simply does not exist... Most computation with groups (especially permutation groups and finitely presented groups) are done with GAP but not all functionalities are accessible through Sage. So you need to learn a bit of GAP.

vdelecroix gravatar imagevdelecroix ( 2013-10-24 16:18:06 -0600 )edit

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Asked: 2013-10-22 15:42:04 -0600

Seen: 118 times

Last updated: Oct 24 '13