Question

Do The same for ${\mathbb{Z}}_{20}$.

Short answer

The list of groups such that every homomorphic image of${\mathbb{Z}}_{20}$is isomorphic to exactly one group on the list is as mentioned below${\mathbb{Z}}_{20},{\mathbb{Z}}_{10},{\mathbb{Z}}_5,{\mathbb{Z}}_4,{\mathbb{Z}}_2,{\mathbb{Z}}_1$

Step-by-step solution

Referring to the result of Exercise 26 in section 7.4

If $G=⟨a⟩$is a cyclic group and $f:G\to H$is a surjective homomorphism of groups, then H is a cyclic group $⟨f\left(a\right)⟩$and $f⟨a⟩$is a generator of H.

Finding the groups

Let us consider $G={\mathbb{Z}}_{20}$.

Since it is a cyclic group with generator 1, so a=$⟨1_{20}⟩$ , and its order is 20.

Now consider a surjective homomorphism $f:{\mathbb{Z}}_{20}\to H$.

From exercise 26, we know that H is also cyclic with generator $f⟨a_{20}⟩$and it must be divisible by 20, because order of${\mathbb{Z}}_{20}$ is 20.

So,$f⟨a_{20}⟩$ has to be from one of the following: 20, 10, 5, 4, 2 and 1.

Therefore, H must be either of .${\mathbb{Z}}_{20},{\mathbb{Z}}_{10},{\mathbb{Z}}_5,{\mathbb{Z}}_4,{\mathbb{Z}}_2,{\mathbb{Z}}_1$