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Let's analyze each of the parts in this question to determine the number of elements in each subgroup:

1. **The subgroup ⟨6, 8⟩ in the group ℤ₂₄:**
   - Here, we are working with the cyclic group ℤ₂₄ under addition modulo 24.
   - The elements of ⟨6, 8⟩ are all integer linear combinations of 6 and 8, reduced modulo 24.
   - This subgroup is generated by the GCD of 6 and 8, which is 2. Therefore, ⟨6, 8⟩ is equivalent to ⟨2⟩ in ℤ₂₄.
   - The order of the subgroup ⟨2⟩ in ℤ₂₄ is \(\frac{24}{2} = 12\).
   - **Answer:** 12 elements.

2. **The subgroup of the quaternions given by ⟨-1, k⟩:**
   - In the quaternion group, \(-1\) and \(k\) are elements.
   - The element \(k\) has order 4, and \(-1\) has order 2.
   - The subgroup generated by \(-1\) and \(k\) includes elements like \(\pm 1\) and \(\pm k\).
   - This forms a subgroup of order 4.
   - **Answer:** 4 elements.

3. **The matrix group ⟨F, R⁴FR²⟩, where \( F = \begin{pmatrix} -1 & -1 \\ 0 & 1 \end{pmatrix} \) and \( R = \begin{pmatrix} 0 & -1 \\ 1 & 1 \end{pmatrix} \):**
   - To determine the order, we need to understand the structure of the group generated by \(F\) and \(R⁴FR²\).
   - Finding exact orders would require matrix multiplication, but typically for small matrix groups, this could represent a finite group of certain symmetries or reflections.
   - More precise computation is needed to find the exact answer.

4. **The subgroup of the dihedral group \( D_{12} \) generated by the reflections \( \sigma_0 \) and \( \sigma_3 \):**
   - The dihedral group \( D_{12} \) has 12 elements, generated by a rotation and a reflection.
   - The reflections \(\sigma_0\) and \(\sigma_3\) are symmetries within this group.
   - The subgroup generated by two reflections generally has 2 elements (identity and the product of two reflections).
   - **Answer:** 2 elements.

Let me know if you'd like detailed calculations, or have any questions on specific parts. Here are five related questions to extend the topic:

1. How do you determine the order of an element in a cyclic group?
2. What is the structure of the quaternion group, and how do we find subgroups within it?
3. How can matrix multiplication help identify elements in a matrix group?
4. What defines the structure of a dihedral group, particularly in terms of reflections and rotations?
5. Why does the GCD of generators help in determining the order of a cyclic subgroup?

**Tip:** When calculating the order of a subgroup in a cyclic group, the order is typically the group's order divided by the GCD of the generators.

For each of the given groups, compute the number of elements: The subgroup ⟨6, 8⟩ in the group ℤ₂₄, the subgroup of the quaternions given by ⟨-1, k⟩, the matrix group ⟨F, R⁴FR²⟩, and the subgroup of the dihedral group D₁₂ generated by the reflections σ₀, σ₃.

Learn how to compute the number of elements in specific subgroups within different groups: ℤ₂₄, the quaternion group, matrix group, and dihedral group D₁₂. This solution explores group theory concepts, including cyclic groups, quaternion structures, and matrix group properties.

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