The number of subgroups in a cyclic group of order 12

A cyclic group is a group that is generated by a single element. In other words, every element in the group can be expressed as a power of a single element, called the generator.

In this case, we are given a cyclic group G of order 12. This means that there exists an element g in G such that the powers of g generate all the elements of G.

Order of a subgroup

The order of a subgroup is the number of elements in the subgroup. In a cyclic group, the order of a subgroup must divide the order of the group.

Since the order of G is 12, the possible orders of subgroups can be 1, 2, 3, 4, 6, or 12.

Number of subgroups

To determine the number of subgroups, we will consider each possible order of the subgroups and count how many subgroups have that order.

1. Subgroups of order 1:
- There is only one subgroup of order 1, which is the trivial subgroup {e}, where e is the identity element.

2. Subgroups of order 2:
- A cyclic group of order 2 has only two elements, the identity element and another element of order 2. This subgroup is isomorphic to the group Z2.
- There is only one subgroup of order 2.

3. Subgroups of order 3:
- A cyclic group of order 3 has only three elements, the identity element and two other elements of order 3. This subgroup is isomorphic to the group Z3.
- There is only one subgroup of order 3.

4. Subgroups of order 4:
- A cyclic group of order 4 has four elements, the identity element and three other elements of order 4. This subgroup is isomorphic to the group Z4.
- There is only one subgroup of order 4.

5. Subgroups of order 6:
- A cyclic group of order 6 has six elements, the identity element and five other elements of order 6. This subgroup is isomorphic to the group Z6.
- There is only one subgroup of order 6.

6. Subgroups of order 12:
- The subgroup of order 12 is the entire group G itself.
- There is only one subgroup of order 12.

Total number of subgroups

Adding up the subgroups of each order, we have:
1 + 1 + 1 + 1 + 1 + 1 = 6

Therefore, the total number of subgroups in the cyclic group G of order 12 is 6.