Find group from matrix presentation

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Mattia Villani

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Mar 23, 2021, 10:00:01 AMMar 23
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Is is possible to find the group given the matrix presentation?

Dima Pasechnik

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Mar 23, 2021, 12:18:12 PMMar 23
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On Tue, Mar 23, 2021 at 2:00 PM Mattia Villani <matt...@gmail.com> wrote:
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> Is is possible to find the group given the matrix presentation?

Please be more specific. Post some Sage commands you're trying.

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Mattia Villani

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Mar 24, 2021, 3:47:09 AMMar 24
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I do not have real code, only a matrix:

matrix([[1,-3],[-3,4]])

which should be a representation of the group Z5: I want to verify it with Sage

Vincent Delecroix

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Mar 24, 2021, 3:50:18 AMMar 24
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Your matrix has determinant 4 - 9 = -5. Hence, the group it generates
in GL(2,QQ) is necessarily infinite.

Mattia Villani

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Mar 24, 2021, 4:28:45 AMMar 24
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That matrix comes from the paper by J.Hempel: "Homology of covering" Pac. J. Math. vol 112 (1984) 83, example 5.2.
The author says that it presents Z5

Vincent Delecroix

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Mar 24, 2021, 4:35:05 AMMar 24
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What does it mean "it presents Z5"?

Mattia Villani

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Mar 24, 2021, 5:01:43 AMMar 24
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I think the author uses "present" to say "generate"

nqn...@gmail.com

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Mar 24, 2021, 5:11:23 AMMar 24
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El miércoles, 24 de marzo de 2021 a las 10:01:43 UTC+1, matt...@gmail.com escribió:
I think the author uses "present" to say "generate"


I suppose you mean the finitely generated Z-module (abelian group) presented by the matrix, ie. its cokernel. In that case, you can do:

sage: A=matrix([[1,-3],[-3,4]])
sage: ZZ^2/A.column_space()
Finitely generated module V/W over Integer Ring with invariants (5)

 

Dima Pasechnik

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Mar 24, 2021, 5:28:46 AMMar 24
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Or just

sage: A.smith_form()
(
[1 0] [ 0 1] [-3 -4]
[0 5], [ 1 -3], [-2 -3]
)

here the 1st entry is the Smith Normal Form of A, and the 2nd and 3rd
ones are what
one uses to conjugate A to get it.

(Yes, in English "present a group" means generators and relations, as
opposed to "represent", which
means a linear representation. Non-native Englsh speakers sometimes
flip the meaning of these, e.g. in Russian
the "представление" means "represenation", and ''копредставление''
means "presentation'')


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