ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 15 Nov 2015 22:44:47 +0100Compute all Elements of a subgroup GL_3(R)https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/ Hi there, i have two matrices A and B, they both out Gl_3(R). I have find all Elements in the Subgroup created by these two. is there a way to do it in sage?
M3 = MatrixSpace(QQ, 3) # Rational numbers
print("Identity matrix:")
show(M3.identity_matrix())
A = M3.matrix([[1, 0, 0], [0, -1, 0], [0, 0, -1]])
B = M3.matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])
There should be 36 Elements in this group B is cyclic B^3=I
and A creates 2 possible Element by negating elements. so the Left inverses create 6 Elements and the Right creates also 6, total is 36. Sun, 15 Nov 2015 14:24:19 +0100https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/Answer by FrédéricC for <p>Hi there, i have two matrices A and B, they both out Gl_3(R). I have find all Elements in the Subgroup created by these two. is there a way to do it in sage? </p>
<p>M3 = MatrixSpace(QQ, 3) # Rational numbers
print("Identity matrix:")
show(M3.identity_matrix())
A = M3.matrix([[1, 0, 0], [0, -1, 0], [0, 0, -1]])
B = M3.matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])</p>
<p>There should be 36 Elements in this group B is cyclic B^3=I
and A creates 2 possible Element by negating elements. so the Left inverses create 6 Elements and the Right creates also 6, total is 36. </p>
https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/?answer=30746#post-id-30746Like that:
sage: M3 = MatrixSpace(QQ, 3)
sage: A = M3.matrix([[1, 0, 0], [0, -1, 0], [0, 0, -1]])
sage: B = M3.matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])
sage: G = MatrixGroup([A,B])
sage: list(G)
[
[1 0 0] [-1 0 0] [0 0 1] [ 0 0 -1] [0 1 0] [ 0 -1 0]
[0 1 0] [ 0 -1 0] [1 0 0] [-1 0 0] [0 0 1] [ 0 0 -1]
[0 0 1], [ 0 0 1], [0 1 0], [ 0 1 0], [1 0 0], [ 1 0 0],
[ 1 0 0] [-1 0 0] [ 0 0 1] [ 0 0 -1] [ 0 1 0] [ 0 -1 0]
[ 0 -1 0] [ 0 1 0] [-1 0 0] [ 1 0 0] [ 0 0 -1] [ 0 0 1]
[ 0 0 -1], [ 0 0 -1], [ 0 -1 0], [ 0 -1 0], [-1 0 0], [-1 0 0]
]
Sun, 15 Nov 2015 19:47:58 +0100https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/?answer=30746#post-id-30746Comment by thetha for <p>Like that:</p>
<pre><code>sage: M3 = MatrixSpace(QQ, 3)
sage: A = M3.matrix([[1, 0, 0], [0, -1, 0], [0, 0, -1]])
sage: B = M3.matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])
sage: G = MatrixGroup([A,B])
sage: list(G)
[
[1 0 0] [-1 0 0] [0 0 1] [ 0 0 -1] [0 1 0] [ 0 -1 0]
[0 1 0] [ 0 -1 0] [1 0 0] [-1 0 0] [0 0 1] [ 0 0 -1]
[0 0 1], [ 0 0 1], [0 1 0], [ 0 1 0], [1 0 0], [ 1 0 0],
[ 1 0 0] [-1 0 0] [ 0 0 1] [ 0 0 -1] [ 0 1 0] [ 0 -1 0]
[ 0 -1 0] [ 0 1 0] [-1 0 0] [ 1 0 0] [ 0 0 -1] [ 0 0 1]
[ 0 0 -1], [ 0 0 -1], [ 0 -1 0], [ 0 -1 0], [-1 0 0], [-1 0 0]
]
</code></pre>
https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/?comment=30749#post-id-30749Thank you, but i have a question. You defined a G as MatrixGroup, could it be defined also with MatrixSubgroup? Since the main Group is GL(3,RR) and the M and N generate only subgroup? Since i would like to calculate the right and the left cosets?Sun, 15 Nov 2015 22:29:45 +0100https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/?comment=30749#post-id-30749Comment by thetha for <p>Like that:</p>
<pre><code>sage: M3 = MatrixSpace(QQ, 3)
sage: A = M3.matrix([[1, 0, 0], [0, -1, 0], [0, 0, -1]])
sage: B = M3.matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])
sage: G = MatrixGroup([A,B])
sage: list(G)
[
[1 0 0] [-1 0 0] [0 0 1] [ 0 0 -1] [0 1 0] [ 0 -1 0]
[0 1 0] [ 0 -1 0] [1 0 0] [-1 0 0] [0 0 1] [ 0 0 -1]
[0 0 1], [ 0 0 1], [0 1 0], [ 0 1 0], [1 0 0], [ 1 0 0],
[ 1 0 0] [-1 0 0] [ 0 0 1] [ 0 0 -1] [ 0 1 0] [ 0 -1 0]
[ 0 -1 0] [ 0 1 0] [-1 0 0] [ 1 0 0] [ 0 0 -1] [ 0 0 1]
[ 0 0 -1], [ 0 0 -1], [ 0 -1 0], [ 0 -1 0], [-1 0 0], [-1 0 0]
]
</code></pre>
https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/?comment=30750#post-id-30750or maybe i am completly on a wrong path?Sun, 15 Nov 2015 22:44:47 +0100https://ask.sagemath.org/question/30739/compute-all-elements-of-a-subgroup-gl_3r/?comment=30750#post-id-30750