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.Fri, 14 Jul 2023 19:10:32 +0200Quotients of multiplicative groups of integers modulo nhttps://ask.sagemath.org/question/70054/quotients-of-multiplicative-groups-of-integers-modulo-n/ 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
240
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! Fri, 14 Jul 2023 14:56:31 +0200https://ask.sagemath.org/question/70054/quotients-of-multiplicative-groups-of-integers-modulo-n/Comment by Max Alekseyev for <p>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:</p>
<pre><code>Zn = Zmod(n)
G = Zn.unit_group()
</code></pre>
<p>Given an element $x$ of $G$, I then create a subgroup generated by an element of G, like so:</p>
<pre><code>H = G.subgroup(x)
</code></pre>
<p>But I can't create a quotient group from G and H:</p>
<pre><code>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
240
241 cdef class AbelianGroup(Group):
</code></pre>
<p>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.</p>
<p>Thank you! </p>
https://ask.sagemath.org/question/70054/quotients-of-multiplicative-groups-of-integers-modulo-n/?comment=70073#post-id-70073Unless there is a better solution, you can convert $G$ and $H$ into permutation groups (like `G.permutation_group()`) and then compute their quotient.Fri, 14 Jul 2023 19:10:32 +0200https://ask.sagemath.org/question/70054/quotients-of-multiplicative-groups-of-integers-modulo-n/?comment=70073#post-id-70073