I have $G=\mathbb{Z}^3$, $H=\langle(3,2,4),(6,1,7),(2,3,6)\rangle \leq G$. My task was to find the quotient $G/H$ as the direct product of infinite cyclic groups and prime-power-order finite groups $(\mathbb{Z}^r \times \mathbb{Z}/n_1\mathbb{Z} \times … \times \mathbb{Z}/n_k\mathbb{Z})$.
Doing it by hand I obtained that $G/H$ is isomorphic to $\mathbb{Z}/25\mathbb{Z}$. However I wanted to check this result with Sage Math, and created these:
G = AdditiveAbelianGroup([0,0,0])
H = G.submodule([(3,2,4),(6,1,7),(2,3,6)])
However doing G.quotient(H)
returned that it was not an implemented function yet.
sage: G.quotient(H)
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-64-ebaeef77a3b4> in <module>
----> 1 G.quotient(H)
/opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/groups/old.pyx in sage.groups.old.Group.quotient (build/cythonized/sage/groups/old.c:2910)()
204 NotImplementedError
205 """
--> 206 raise NotImplementedError
207
208 cdef class AbelianGroup(Group):
NotImplementedError:
sage:
The fact that I can only find documentation for quotients of multiplicative abelian groups is discouraging. If this is not it, what can be the way to find and thus check quotients like these in Sage?
Best Answer
Welcome to MSE!
If you look at the documentation for additive abelian groups, it says
This is a clue that, if things aren't working, we should check out the
fg_pid
module instead. Going to the documentation there, we see an example which is easily adapted to your problem:When you run
G/H
above, you should get the print outFinitely generated module V/W over Integer Ring with invariants (25)
which tells you that $G/H \cong \mathbb{Z} / (25)$, agreeing with your computation.
I hope this helps ^_^