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.