Write the Cayley table for the Dihedral group D4 with 8 elements.
Why is D8 not isomorphic to Z/8Z?
The Correct Answer and Explanation is :
Cayley Table for the Dihedral Group D4D_4:
The Dihedral group D4D_4 represents the symmetries of a square and has 8 elements: four rotations and four reflections. The elements of D4D_4 can be represented as:
- r0r^0 (the identity rotation),
- r1r^1, r2r^2, r3r^3 (rotations by 90°, 180°, and 270° respectively),
- s1s^1, s2s^2, s3s^3, s4s^4 (reflections across axes).
The Cayley table for D4D_4 is as follows, with the elements r0,r1,r2,r3r^0, r^1, r^2, r^3 for rotations and s1,s2,s3,s4s^1, s^2, s^3, s^4 for reflections:
| * | r0r^0 | r1r^1 | r2r^2 | r3r^3 | s1s^1 | s2s^2 | s3s^3 | s4s^4 |
|---|---|---|---|---|---|---|---|---|
| r0r^0 | r0r^0 | r1r^1 | r2r^2 | r3r^3 | s1s^1 | s2s^2 | s3s^3 | s4s^4 |
| r1r^1 | r1r^1 | r2r^2 | r3r^3 | r0r^0 | s2s^2 | s3s^3 | s4s^4 | s1s^1 |
| r2r^2 | r2r^2 | r3r^3 | r0r^0 | r1r^1 | s3s^3 | s4s^4 | s1s^1 | s2s^2 |
| r3r^3 | r3r^3 | r0r^0 | r1r^1 | r2r^2 | s4s^4 | s1s^1 | s2s^2 | s3s^3 |
| s1s^1 | s1s^1 | s4s^4 | s3s^3 | s2s^2 | r1r^1 | r2r^2 | r3r^3 | r0r^0 |
| s2s^2 | s2s^2 | s3s^3 | s4s^4 | s1s^1 | r2r^2 | r3r^3 | r0r^0 | r1r^1 |
| s3s^3 | s3s^3 | s2s^2 | s1s^1 | s4s^4 | r3r^3 | r0r^0 | r1r^1 | r2r^2 |
| s4s^4 | s4s^4 | s1s^1 | s2s^2 | s3s^3 | r0r^0 | r1r^1 | r2r^2 | r3r^3 |
Why is D8D_8 Not Isomorphic to Z/8Z\mathbb{Z}/8\mathbb{Z}?
D8D_8, the Dihedral group of order 8, and Z/8Z\mathbb{Z}/8\mathbb{Z}, the cyclic group of order 8, are not isomorphic due to their structural differences. The two key reasons for this are:
- Cyclic vs. Non-Cyclic Structure:
- Z/8Z\mathbb{Z}/8\mathbb{Z} is a cyclic group. This means that there exists an element, specifically the generator 1, such that every other element in the group can be written as a power of this generator. In other words, Z/8Z\mathbb{Z}/8\mathbb{Z} has an element of order 8, which generates the entire group.
- D8D_8, on the other hand, is not cyclic. It is generated by two elements: a rotation (say, rr) and a reflection (say, ss). The rotation elements in D8D_8 only generate a subgroup of order 4, while the reflections generate a subgroup of order 2. There is no element in D8D_8 that has order 8. This non-cyclic nature of D8D_8 directly implies that it cannot be isomorphic to the cyclic group Z/8Z\mathbb{Z}/8\mathbb{Z}.
- Element Orders:
- In Z/8Z\mathbb{Z}/8\mathbb{Z}, the element 1 has order 8, meaning that repeated addition of 1 (mod 8) will eventually cover all elements of the group.
- In D8D_8, the highest order of any element is 4 (from the rotations). The elements in D8D_8 have orders 1, 2, and 4, and there is no element with order 8, unlike in Z/8Z\mathbb{Z}/8\mathbb{Z}.
Therefore, the structural differences in element orders and the cyclic versus non-cyclic nature of these groups prevent D8D_8 from being isomorphic to Z/8Z\mathbb{Z}/8\mathbb{Z}.