Finding minimal relations for a group generated by matrices

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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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1 Answer

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answered 0 years ago

Max Alekseyev

6922 ●9 ●52 ●161

updated 0 years ago

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?