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.