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. ....
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.
Right. Dan Asimov <dasimov@earthlink.net> wrote:
I think it would also fair to say {0} is a group under multiplication modulo 10.
Right. And that's all eight of them. Mod N, the identity elements are those numbers I such that I^2 = I mod N. For N=10, those are 0, 1, 5, and 6. The numbers of identity elements mod N appear to be given by A034444. Mod 30, there are 8 of them, and 28 groups. Every number 0...N-1 "belongs to" at most one identity element. For N=10, 0 belongs to 0; 1,3,5,7 belong to 1; 5 belongs to 5; and 2,4,6,8 belong to 6. That's all 10 numbers. They each appear in at least one group. Mod N, the number of numbers that *don't* appear in any multiplicative group appears to be given by A055654. It appears to be always 0 for square-free N, and never 0 if N is not square free. Those groupless numbers are always divisors of N. I think the order of any of these groups always divides phi(N). Except for very small N, you'll always have {0}, {1}, {1, N-1}, and the phi(N) numbers relatively prime to N. The last may have other subgroups. If M divides N, I think you'll always also have, for each group in M, the same group in N only with each term multiplied by N/M. For instance {2,4,6,8} where N=10 is an echo of {1,2,3,4} where N=5. And {5} where N=10 is an echo of {1} where N=2. So there's probably some simple recurrence relation which can be used to find how many multiplicative groups there are mod N. How many are there? 1,2,3,3,4,6,5,6,5,8,5,10,7,10,14,9,6,10,7,14,17,10,5,24,7,14,7,17,7,28,9,12,17,12,24,17,... Not in OEIS. I plan to add it. This discussion thread makes it obvious that I need help in phrasing my description of what this is about, so someone please help. Thanks.