I haven't been reading all these messages, but here goes anyway.... In every finite commutative monoid M, you get its maximal subgroups by.... (1) Computing first its set of idempotents X (ie those elements x such that x^2 =x). (2) Then, for each x in X, compute the subset S(x) of y's such that x divides y and y divides x. Each set S(x) is a maximal subgroup of M with identity the idempotent x. For the particular case of the integers modulo 30, I got these idempotents and maximal subgroups In[1164]:= idempotents = Select[elements, mult[#, #] == # &] Out[1164]= {0, 1, 6, 10, 15, 16, 21, 25} Ie, there are 8 idempotents. So there will be 8 maximal subgroups. Here they are. In[1173]:= Map[allMutuals, idempotents] Out[1173]= {{0}, {1, 7, 11, 13, 17, 19, 23, 29}, {6, 12, 18, 24}, {10, 20}, {15}, {2, 4, 8, 14, 16, 22, 26, 28}, {3, 9, 21, 27}, {5, 25}} Listing all the subgroups is more work, but they're easily found via this divide and conquer technique. On Fri, Apr 22, 2016 at 10:34 AM, Tom Rokicki <rokicki@gmail.com> wrote:
{1, 7, 13, 19} is not a group; 7 * 13 = 11.I haven't checked the rest of your list.
My wetware says (7*13)%30 = (91 % 30) = 1.
This is not a big deal if you're generating the groups by hand, but if you're doing it programatically, it means there's something wrong with your program.
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