Quotients of multiplicative groups of integers modulo n

asked 2023-07-14 14:56:31 +0200

eluoss gravatar image

I am studying multiplicative groups of integers modulo $n$; in particular, this is the group $G=\mathbb{Z}_n^\ast$. I am particularly interested in studying the structure of quotient groups of the form $G=\mathbb{Z}_n^\ast/\langle p \rangle$. I know that I can work with finite Abelian groups as follows:

Zn = Zmod(n)
G = Zn.unit_group()

Given an element $x$ of $G$, I then create a subgroup generated by an element of G, like so:

H = G.subgroup(x)

But I can't create a quotient group from G and H:

Q = G.quotient(H)
File /private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/groups/group.pyx:239, in sage.groups.group.Group.quotient (build/cythonized/sage/groups/group.c:3453)()
    237             NotImplementedError
    238         """
--> 239         raise NotImplementedError
    241 cdef class AbelianGroup(Group):

Is there any other way I can work with quotient groups of multiplicative Abelian groups? I'm trying to peek at their structure (we know that any finite Abelian group can be written as the direct product of cyclic groups). I can probably implement multiplicative quotient groups myself but I'd rather not do this of course.

Thank you!

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Unless there is a better solution, you can convert $G$ and $H$ into permutation groups (like G.permutation_group()) and then compute their quotient.

Max Alekseyev gravatar imageMax Alekseyev ( 2023-07-14 19:10:32 +0200 )edit