# Compute all Elements of a subgroup GL_3(R)

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.

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Like 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]
]

more

Thank 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?

( 2015-11-15 15:29:45 -0600 )edit

or maybe i am completly on a wrong path?

( 2015-11-15 15:44:47 -0600 )edit