Question

Let G be a group of order 12. Which of the following is a possible order of a subgroup of G ?

Question 9Answer

a.
7


b.
8


c.
6


d.
5

Answer 1

To determine the possible orders of subgroups of a group \( G \) of order 12, we can use Lagrange's theorem. This theorem states that the order of any subgroup of a finite group \( G \) must divide the order of \( G \).

The order of the group \( G \) is 12. The divisors of 12 are:

- 1
- 2
- 3
- 4
- 6
- 12

Thus, the possible orders for subgroups of \( G \) must be one of these divisors.

Now, let's analyze the options provided:

a. **7**: This does not divide 12, so it's not a possible order of a subgroup.

b. **8**: This does not divide 12, so it's not a possible order of a subgroup.

c. **6**: This divides 12, so it is a possible order of a subgroup.

d. **5**: This does not divide 12, so it's not a possible order of a subgroup.

Therefore, the only option that is a possible order of a subgroup of \( G \) is:

**c. 6**