Wait, I think I just realized. Consider, for instance, {2,4,6,8}. Under multiplication, these do form a group with identity 6. I didn't think of it because it doesn't "cohere" with the ring of integers mod 10. {5} is another example. I'm not sure how to enumerate these. {6} and {4,6} are two more. On Wed, Apr 20, 2016 at 9:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Modulo 10, only 1, 3, 7, and 9 have reciprocals. These four form a cyclic group under multiplication, generated by 3 (or 7). This group has only two subgroups, {1} and {1,9}. These are the three groups Dan enumerated. I don't think there are any others; so perhaps I still haven't understood what Keith is trying to enumerate. Keith, can you give another example of a subset of the integers modulo 10 that meets your criteria?
On Wed, Apr 20, 2016 at 8:39 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
Dan Asimov <asimov@msri.org> wrote:
Keith F. Lynch <kfl@KeithLynch.net> wrote:
List all multiplicative groups of integers mod ten. For instance {1,9} is one of them, and {1} is another.
If I understand the problem, it's to list all subgroups of the group of invertible elements of the ring Z/10.
The invertible elements can be chosen as {-3, -1, 1, 3}.
Since there is an element of order 4, this is isomorphic to the abelian group Z/4, so it has: ....
{-3, -1, 1, 3}, {-1, 1}, {1}.
Not quite what I meant. Yes, those are solutions, but those are not the only solutions.
I meant any subset of {0,1,2,3,4,5,6,7,8,9} which forms a group under multiplication modulo 10. There are of course 1024 subsets, and I brute-force tested every one of them just for lulz. Similarly with all other moduli 1 through 36.
How can I phrase that better? Thanks.
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