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Finding minimal relations for a group generated by matrices

asked 2024-12-30 22:38:31 +0100

sagequstions gravatar image

I have a group G, which is generated by two $n \times n$ matrices with integer entries. For example:

C=matrix([[0,0,-1],[1,0,-1],[0,1,-1]])
R=matrix([[0,0,1],[1,0,0],[0,1,0]])
G = MatrixGroup([C,R])
display(G.as_permutation_group().structure_description())

My question is: How can I obtain a group isomorphism from a free group F in two variables x,y with relations I to G sending x to C and y to R? So Im interested in getting minimal relations in the two generators C and R that describe the group G using Sage. I looked at https://doc.sagemath.org/html/en/refe... but it seems that the group has to be already in the form of a free group modulo relations to use most commands there.

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answered 2024-12-31 09:17:42 +0100

Max Alekseyev gravatar image

updated 2024-12-31 09:19:05 +0100

Perhaps, you want:

G.as_permutation_group().as_finitely_presented_group(reduced=True)
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Thank you. I know that in my problem the matrices C and R always have finite order. So I want in the minimal relations the relations C^n=1 and R^m=1. Is there a way to force to have those two relations in the minimal relations?

sagequstions gravatar imagesagequstions ( 2024-12-31 10:56:53 +0100 )edit

The reduced here means that no relations is unnecessary (meaning implied by the other relations), right?

sagequstions gravatar imagesagequstions ( 2024-12-31 11:00:55 +0100 )edit

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Asked: 2024-12-30 22:38:31 +0100

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Last updated: Dec 31 '24