SOLUTION: Find all the orders of subroups of Z20? Hi The Best Teachers online! Can you help me on this question please? Thank you! God bless!

For any subgroup of a group, its order divides the order of the group.


So, we potentially have the orders of subgroups as 2, 4, 5 and 10  (all the divisors of the number 20).


Now, each of these numbers is really realized as the order of a subgroup.


For it, you should know that Z20 is the additive group of integers from 0 to 19 modulus 20.


Then the subgroup of the order 2 is the subgroup (the subset) {0,10} of Z20;

     the subgroup of the order 4 is the subgroup (the subset) {0,5,10,15} of Z20;

     the subgroup of the order 5 is the subgroup (the subset) {0,4,8,12,16} of Z20;

     the subgroup of the order 10 is the subgroup (the subset) {0,2,4,6,8,10,12,14,16,18} of Z20.