Check if a finitely generated matrix group is finite (works with QQ and not with CC)

asked 2017-07-25 17:22:22 +0100

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Dear all, I am a newbie in sage. I would like to check if a finitely generated matrix group is finite. Before to proceed with the calculation on my actual problem (where matrices have complex entries), I have tried a very simple example. Consider the group generated by the matrices [1,0,0,1] and [0,1,1,0], this group is clearly finite. Can somebody explain me why the following code works:

sage: MS = MatrixSpace(QQ, 2, 2)
sage: G = MatrixGroup([MS([1,0,0,1]),MS([0,1,1,0])])
sage: G.is_finite()
True

but if I change the field QQ -> RR (or CC), an error is generated:

sage: MS = MatrixSpace(RR, 2, 2)
sage: G = MatrixGroup([MS([1,0,0,1]),MS([0,1,1,0])])
sage: G.is_finite()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-215-0022a668c150> in <module>()
----> 1 G.is_finite()

/Applications/SageMath-7.6.app/Contents/Resources/sage/src/sage/groups/group.pyx in sage.groups.group.Group.is_finite (/Applications/SageMath-7.6.app/Contents/Resources/sage/src/build/cythonized/sage/groups/group.c:2696)()
179             NotImplementedError
180         """
--> 181         return self.order() != infinity
182 
183     def is_multiplicative(self):

/Applications/SageMath-7.6.app/Contents/Resources/sage/src/sage/groups/group.pyx in sage.groups.group.Group.order (/Applications/SageMath-7.6.app/Contents/Resources/sage/src/build/cythonized/sage/groups/group.c:2623)()
164             NotImplementedError
165         """
--> 166         raise NotImplementedError
167 
168     def is_finite(self):

NotImplementedError:

Is there any way to force the second piece of code to work with matrices with entries in CC?

Thank you in advance.

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In the first case we can also compute the order, and the list of elements. In the second case, we use non-exact objects. (Why do we do this!?) Then i expect humanly that there is no multiplicative order method implemented for such a numerical matrix, being really glad that there is no one! Why should we work with inexact entries in a finite matrix group?

Note that we run into a NotImplementedError: multiplicative order is only implemented for matrices over finite fields in both following cases...

g = matrix( RR, 2, 2, [ sqrt(3)/2.0, 0.5, 0.5, sqrt(3)/2.0 ] )
g.multiplicative_order()    # --> NotImplementedError

# and

sage: K.<a> = QuadraticField(3)
sage: g = matrix(K, 2,2, [a/2,1/2,-1/2,a/2] )
sage: g^12
[1 0]
[0 1]
sage: g.multiplicative_order()    # --> NotImplementedError

dan_fulea ( 2017-07-25 20:32:20 +0100 )edit

Use an exact ring, such as QQbar or AA or UniversalCyclotomicField instead of CC.

FrédéricC ( 2017-07-27 08:13:06 +0100 )edit

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Check if a finitely generated matrix group is finite (works with QQ and not with CC)

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