Number of proper Subgroups of Z6

The group Z6 is the set of integers modulo 6, which is {0, 1, 2, 3, 4, 5}. To determine the number of proper subgroups of Z6, we need to find all the subsets of Z6 that form subgroups.

To form a subgroup, the subset must satisfy three conditions:
1. It must contain the identity element, which is 0 in Z6.
2. It must be closed under addition and subtraction.
3. It must contain the additive inverse of each element.

Let's consider each possible subset and check if it forms a subgroup:

1. {0} - This subset contains only the identity element, and it is closed under addition and subtraction. Therefore, it forms a subgroup.

2. {0, 3} - This subset contains the identity element and the element 3. It is closed under addition and subtraction, and it contains the additive inverse of each element. Therefore, it forms a subgroup.

3. {0, 2, 4} - This subset contains the identity element and the elements 2 and 4. It is closed under addition and subtraction, and it contains the additive inverse of each element. Therefore, it forms a subgroup.

4. {0, 1, 2, 3, 4, 5} - This subset contains all the elements of Z6. It is closed under addition and subtraction, and it contains the additive inverse of each element. However, it does not form a proper subgroup because it is equal to the entire group Z6.

Therefore, there are 3 proper subgroups of Z6.

In summary, the correct answer is option 'A' - 3.