# Order of Elements in the Dihedral Group \( D_8 \) The dihedral group \( D_8 \) is the group of symmetries of a regular octagon, including rotations and reflections. It has 8 elements. ## Step 1: Presentation of \( D_8 \) The group \( D_8 \) can be presented as: \[ D_8 = \langle r, s \mid r^4 = s^2 = 1,\, sr = r^{-1}s \rangle \] where: - \( r \) is a rotation by \( 90^\circ \) (i.e., \( 360^\circ/4 \)), - \( s \) is a reflection. ## Step 2: List All Elements The elements of \( D_8 \) are: \[ \{ 1,\, r,\, r^2,\, r^3,\, s,\, sr,\, sr^2,\, sr^3 \} \] ## Step 3: Find the Order of Each Element ### **Rotations** 1. **Identity**: \( 1 \) - Order: 1 2. **Rotation by \( 90^\circ \)**: \( r \) - Order: \( 4 \) (since \( r^4 = 1 \), but smaller powers are not 1) 3. **Rotation by \( 180^\circ \)**: \( r^2 \) - Order: \( 2 \) (since \( (r^2)^2 = r^4 = 1 \)) 4. **Rotation by \( 270^\circ \)**: \( r^3 \) - Order: \( 4 \) (since \( (r^3)^4 = r^{12} = (r^4)^3 = 1^3 = 1 \), and lower powers are not 1) ### **Reflections** All reflections are of the form \( sr^k \) for \( k = 0,1,2,3 \): 5. **Reflection \( s \)** - Order: 2 (since \( s^2 = 1 \)) 6. **Reflection \( sr \)** - Order: 2 (since \( (sr)^2 = srsr = s(r s)r = s(s r^{-1})r = (s^2) r^{-1} r = 1 \)) 7. **Reflection \( sr^2 \)** - Order: 2 (same computation as above) 8. **Reflection \( sr^3 \)** - Order: 2 (same computation as above) ## Step 4: List the Orders | Element | Order | |-----------|-------| | \( 1 \) | 1 | | \( r \) | 4 | | \( r^2 \) | 2 | | \( r^3 \) | 4 | | \( s \) | 2 | | \( sr \) | 2 | | \( sr^2 \) | 2 | | \( sr^3 \) | 2 | --- ## **Final Answer** The orders of elements in \( D_8 \) are: - **Order 1:** \( 1 \) - **Order 2:** \( r^2, s, sr, sr^2, sr^3 \) - **Order 4:** \( r, r^3 \) **Summary Table:** | Element | Order | |-----------------|-------| | \( 1 \) | 1 | | \( r \) | 4 | | \( r^2 \) | 2 | | \( r^3 \) | 4 | | \( s \) | 2 | | \( sr \) | 2 | | \( sr^2 \) | 2 | | \( sr^3 \) | 2 |