Can anyone explain how to do this problem. I understand that since this has an order of 9 then it is generated by 6, but I just don't know how to find the subgroups. Thanks!
--Lchinn 22:03, 23 February 2009 (UTC)
I don't think that there is a subgroup of order 9. I think because the lcm of the orders can not be equal to 9, therefore, there is no subgroup of order 9. --Awika 12:39, 24 February 2009 (UTC)
The lcm can be equal 9:
The subgroups have the form (a,b,c) = H. a is in Z12, b is in Z4 and c is in Z15. By theorem 8.1 in order for |H| = 9, lcm(|a|,|b|,|c|) = 9, thus |a|, |b|, |c| = 1, 3 or 9, who's lcm is 9. Thus, find a value of a, b and c that this statement is true, meaning that there is a representative of orders 1, 3 and 9 present.
ex. a = 8 (|4| = 9), b = 0 (|0| = 1), c = 5 (|5| = 3) => H = (8,0,5), under addition, 9*H = (72,0,45) = (0,0,0).
-K. Brumbaugh
Isn't |4|=3 in Z12, same with |8|? I don't see how you can find an element of order 9 in Z12.-Paul
