No of homomorphism from D4 to Q8?

Question

Answer

Introduction

To find the number of homomorphisms from the dihedral group D4 to the quaternion group Q8, we need to understand the properties and structures of these groups.

Dihedral group D4

The dihedral group D4 consists of the symmetries of a square. It has 8 elements, which can be represented by the following permutations:
D4 = {e, r, r^2, r^3, f, fr, fr^2, fr^3}
where e represents the identity, r represents a 90-degree rotation, and f represents a reflection.

Quaternion group Q8

The quaternion group Q8 consists of 8 elements, which can be represented by the following quaternion units:
Q8 = {1, -1, i, -i, j, -j, k, -k}
where 1 is the identity element, -1 is the negative of 1, and i, j, k are imaginary units with the properties i^2 = j^2 = k^2 = -1.

Homomorphisms

A homomorphism is a function between two groups that preserves the group structure. In other words, if φ: G → H is a homomorphism, then for any elements a, b in G, we have φ(ab) = φ(a)φ(b).

To find the number of homomorphisms from D4 to Q8, we need to consider the possible mappings between the elements of D4 and Q8.

Possible Mappings

Let's consider the elements of D4 and their possible mappings to Q8:

1. Identity element: The identity element of D4 must be mapped to the identity element of Q8, which is 1.

2. Rotation elements: The rotation elements of D4 (r, r^2, r^3) must be mapped to the powers of i in Q8. Since i^2 = -1, we have the following mappings:
- r → i
- r^2 → -1
- r^3 → -i

3. Reflection elements: The reflection elements of D4 (f, fr, fr^2, fr^3) must be mapped to the negative of the quaternion units in Q8. We have the following mappings:
- f → -1
- fr → -i
- fr^2 → 1
- fr^3 → i

Conclusion

In summary, there are 8 possible homomorphisms from the dihedral group D4 to the quaternion group Q8, as each element of D4 can be mapped to a unique element in Q8. These mappings preserve the group structure and satisfy the homomorphism property.

Answer

Number of Homomorphisms from D4 to Q8

To determine the number of homomorphisms from the dihedral group D4 to the quaternion group Q8, we need to analyze the structure of both groups and the properties of homomorphisms.

Diheral Group D4

The dihedral group D4 is the group of symmetries of a square. It has 8 elements and is generated by two elements: a rotation R of 90 degrees and a reflection F along a diagonal. The group multiplication table for D4 is as follows:

```
| e | R | R^2 | R^3 | F | FR | FR^2 | FR^3 |
----------------------------------------------------
e | e | R | R^2 | R^3 | F | FR | FR^2 | FR^3 |
----------------------------------------------------
R | R | R^2 | R^3 | e | FR | FR^2 | FR^3 | F |
----------------------------------------------------
R^2| R^2 | R^3 | e | R | FR^2 | FR^3 | F | FR |
----------------------------------------------------
R^3| R^3 | e | R | R^2 | FR^3 | F | FR | FR^2 |
----------------------------------------------------
F | F | FR | FR^2| FR^3| e | R | R^2 | R^3 |
----------------------------------------------------
FR | FR | FR^2| FR^3| F | R | R^2 | R^3 | e |
----------------------------------------------------
FR^2|FR^2| FR^3| F | FR | R^2 | R^3 | e | R |
----------------------------------------------------
FR^3|FR^3| F | FR | FR^2| R^3 | e | R | R^2 |
```

Quaternion Group Q8

The quaternion group Q8 is a non-abelian group of order 8. It can be represented by the following elements: {1, -1, i, -i, j, -j, k, -k}, where the multiplication table for Q8 is given by:

```
| 1 | -1 | i | -i | j | -j | k | -k |
--------------------------------------------------
1 | 1 | -1 | i | -i | j | -j | k | -k |
--------------------------------------------------
-1 | -1 | 1 | -i | i | -j | j | -k | k |
--------------------------------------------------
i | i | -i | -1 | 1 | k | -k | -j | j |
--------------------------------------------------
-i | -i | i | 1 | -1 | -k |

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