Let H be a subgroup of order 8 of  the integer group 24. List the elements of H.

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.

Explore how to find elements of a subgroup of order 8 in the group Z24. Understand the concepts of subgroups, cyclic groups, and modular arithmetic. Key theorem used: Lagrange's Theorem. Suitable for advanced undergraduate students and beyond.

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