Determine the commutator subgroup of D8 and Q8

To determine the commutator subgroup of the dihedral group \( D_8 \) and the quaternion group \( Q_8 \), let’s go through each case.

### 1. Commutator Subgroup of \( D_8 \)
The dihedral group \( D_8 \) represents the group of symmetries of a square, including rotations and reflections, and has the following properties:

- **Order**: 8
- **Elements**: \( D_8 = \{ e, r, r^2, r^3, s, rs, r^2s, r^3s \} \)
  - Here, \( r \) represents a rotation by \( 90^\circ \) and \( s \) represents a reflection.

To find the commutator subgroup \( D_8' \), we calculate the set of all commutators in \( D_8 \):
\[
[x, y] = x^{-1} y^{-1} xy
\]
### Commutator Calculations for \( D_8 \):
1. For **rotations** \( r \), \( r^2 \), and \( r^3 \), the commutators are trivial (i.e., they commute with each other).
2. For reflections like \( s \) and products like \( rs \), we compute specific commutators:
   - \( [r, s] = r^{-1} s^{-1} r s = r^3 s r s = r^2 \)
   - Similarly, other commutators yield elements like \( r^2 \).

After computing all commutators, we find that \( D_8' = \langle r^2 \rangle = \{ e, r^2 \} \), which is the **cyclic subgroup generated by \( r^2 \)**.

Thus, the **commutator subgroup of \( D_8 \) is \( \{ e, r^2 \} \)**.

---

### 2. Commutator Subgroup of \( Q_8 \)
The quaternion group \( Q_8 \) has the following elements:

- **Order**: 8
- **Elements**: \( Q_8 = \{ \pm 1, \pm i, \pm j, \pm k \} \)
  - Here, \( i, j, k \) are the quaternion units, satisfying \( i^2 = j^2 = k^2 = -1 \) and \( ij = k \), \( jk = i \), \( ki = j \).

For \( Q_8 \), we compute the commutators in a similar way:
- **Commutator** \( [i, j] = i^{-1} j^{-1} i j = (-i)(-j)ij = k^2 = -1 \)
- Similarly, all non-identity commutators in \( Q_8 \) yield \( -1 \).

Thus, the commutator subgroup \( Q_8' \) is the subgroup generated by \( -1 \):
\[
Q_8' = \langle -1 \rangle = \{ 1, -1 \}
\]

Therefore, the **commutator subgroup of \( Q_8 \) is \( \{ 1, -1 \} \)**.

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### Summary
- **Commutator subgroup of \( D_8 \):** \( \{ e, r^2 \} \)
- **Commutator subgroup of \( Q_8 \):** \( \{ 1, -1 \} \)

Would you like further details on commutator subgroup derivations or specific steps?

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1. What is the significance of the commutator subgroup in group theory?
2. How does the structure of \( D_8 \) compare to other dihedral groups?
3. Why is the commutator subgroup of \( Q_8 \) related to the element \( -1 \)?
4. Can we generalize commutator subgroups for other quaternion groups?
5. What other properties distinguish \( D_8 \) from \( Q_8 \)?

**Tip:** In group theory, the commutator subgroup is always a normal subgroup and often reveals information about a group’s structure, such as whether it is abelian.

Explore the commutator subgroups of the dihedral group D8 and quaternion group Q8, including step-by-step calculations and theorems. Learn about group theory concepts like commutator subgroups and the significance of normal subgroups.

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