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How to Calculate the Subgroups of Zn

The cyclic groups are a subset of all groups with a particularly easy-to-understand structure. In particular, the cyclic groups can be represented by a set of numbers with modulo arithmetic. For example, Z15 can be formed by the numbers 0 to 14, with 16 equal to 1, 17 equal to 2 and so on. These cyclic groups have a mathematics all their own. One particularly interesting question, which yields deep insights in undergraduate math classes, is what subsets of these groups form groups themselves.

Instructions

- 1
  Factor the order of your group. For example, if the group has 18 elements, its order is 18: 18 = 2 x 3 x 3. If the group has 30 elements, its order is 30: 2 x 3 x 5.
- 2
  Determine all possible numbers that can divide evenly into the order of the group, based on the factorization done in Step 1. In a group of order 18, this would give 2, 3, 6 and 9. In a group of order 30, this gives 2, 3, 5, 6, 10 and 15.
- 3
  Understand that every subgroup of your cyclic group must be of the order of a factor of your main group's order. For example, for the cyclic group of order 18, a proper subgroup --- or a subgroup that is larger than one element and smaller than 18 elements --- must be of order 2, 3, 6 or 9, since these are the only numbers that can factor into 18. Additionally, every subgroup of a subgroup of a cyclic group must itself be a cyclic group.
- 4
  Find the smallest element of each of the numbers found in Step 2. In the group of order 18 under addition, 2 is the smallest element of order 9 (since 2+2+2+2+2+2+2+2+2=18), 3 is the smallest element of order 6 (since 3+3+3+3+3+3=18), 6 is the smallest element of order 3 (since 6+6+6=18) and 9 is the smallest element of order 2 (since 9+9=18).
- 5
  Determine the subgroups formed by these elements. In the cyclic group of order 18, the subgroup generated by 2 is the group {0, 2, 4, 6, 8, 10, 12, 14, 16}. The subgroup generated by 3 is the group {0, 3, 6, 9, 12, 15}, and that generated by 6 is {0, 6, 12}. The cyclic subgroup of order 2 is the group {0, 9}. Thanks to the combination of properties discussed in Step 3, there is always exactly one subgroup of a cyclic group for each number that can divide evenly into the order of the group.

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