Question

Find all homomorphic images of $D_4$.

Short answer

The homomorphic images of $D_4$are$\left\{0\right\},{\mathbb{Z}}_2,{\mathbb{Z}}_2\times {\mathbb{Z}}_2$ and $D_4$.

Step-by-step solution

Referring to the First IsomorphismTheorem

Theorem 8.20

Let$f:G\to H$ be a surjective homomorphism of a group with kernel K. Then the quotient group$G/K$ is isomorphic to H.

Finding the homomorphic images of D4

Consider a map $f:D_4\to G$.

Then, byFirst Isomorphism Theorem, $D_4/K\cong G$.

The order of$D_4/K$= .$\left|D_4\right|/\left|K\right|$

As we know that $\left|D_4\right|$=8, the order of the image must be a divisor of 8.

So it can be 1, 2, 4, or 8.

Now, in order to find a homomorphic image, we need to check every possible case of the homomorphic image with the given orders:

$f:D_4\to G$

1) Case 1 - Suppose the order is 1.

Then, it represents a trivial group.

Therefore, $f:D_4\to \left\{0\right\}$.

By ,$f\left(a\right)=0$for .$a\in D_4$

So,$\left\{0\right\}$is an homomorphic image of $D_4$.

2) Case 2 - Suppose the order is 2.

Considering $N=\{d,h,t,v\}$.

We know that, if N is a normal subgroup of $D_4$then ${D_4/N\cong \mathbb{Z}}_2$.

Therefore, $f:D_4\to {\mathbb{Z}}_2$, and $f\left(a\right)$can be given as:

$f\left(a\right)=\{\begin{array}{l}0ifa=d,h,t,v\\ 1ifa=r_0,r_1,r_2,r_3\end{array}$

3) Case 3 - Suppose the order is 4.

We know that every group of order 4 is isomorphic to with.${\mathbb{Z}}_{4}or{\mathbb{Z}}_2\times {\mathbb{Z}}_2$

Let’s take$N=\{r_0,r_1\}$.

If N is the normal group then${D_4/N\cong \mathbb{Z}}_2\times {\mathbb{Z}}_2$.

Therefore $f:D_4\to {\mathbb{Z}}_2\times {\mathbb{Z}}_2$,and$f\left(a\right)$can be given as:

$f\left(a\right)=\left\{\begin{array}{l}(0,0)ifa=r_0{r}_1\\ (1,0)ifa=r_1{r}_2\\ (0,1)ifa=h,v\\ (1,1)ifa=d,t\end{array}\right\}$

4) Case 4 - Suppose the order is 8.

Consider$f:D_4\to D_4$ .

Here, the kernel is trivial because both the groups have the same order.

So,$D_4$is homomorphic image of$D_4$ .

Therefore, the homomorphic images of $D_4$are $\left\{0\right\},{\mathbb{Z}}_2,{\mathbb{Z}}_2\times {\mathbb{Z}}_2$and $D_4$.