In this problem, we need to find a subgroup $H$ of order 8 in the group $\mathbb{Z}_{24}$, which is the group of integers modulo 24 under addition.

Step 1: Understanding $\mathbb{Z}_{24}$

The group $\mathbb{Z}_{24}$ consists of the elements: $\mathbb{Z}_{24} = \{0, 1, 2, 3, \ldots, 23\}$ with addition performed modulo 24.

Step 2: Identifying Subgroups

The order of a subgroup $H$ in $\mathbb{Z}_{24}$ must divide the order of the group $\mathbb{Z}_{24}$, which is 24. Thus, a subgroup of order 8 is possible.

Subgroups in $\mathbb{Z}_{24}$ are generated by elements whose orders divide 24. To have a subgroup of order 8, we need an element $g$ whose order is 3, since $24/3 = 8$.

Step 3: Generating Subgroup of Order 8

The element of order 3 that generates a subgroup of order 8 is $g = 3$.

The elements of the subgroup generated by $3$ in $\mathbb{Z}_{24}$ are: $H = \langle 3 \rangle = \{0, 3, 6, 9, 12, 15, 18, 21\}$

Answer:

The elements of the subgroup $H$ of order 8 in $\mathbb{Z}_{24}$ are: $\{0, 3, 6, 9, 12, 15, 18, 21\}$

Do you want details or have any questions?

Here are some related questions:

1. How do you find the order of an element in a cyclic group?
2. Can you list all subgroups of $\mathbb{Z}_{24}$?
3. What is the significance of the greatest common divisor in finding subgroups of cyclic groups?
4. How many distinct subgroups of order 8 are there in $\mathbb{Z}_{24}$?
5. What is the connection between cyclic subgroups and generators?
6. What are cosets in the context of the group $\mathbb{Z}_{24}$?
7. How do we prove that $H = \langle 3 \rangle$ is indeed a subgroup of order 8?
8. What is the role of Lagrange's Theorem in determining subgroup orders?

Tip: In cyclic groups, any subgroup is also cyclic, and its order is a divisor of the group’s order.