How many group homomorphisms are there from $\Bbb Z_{20}$ to $\Bbb Z_{10}$?

Question

Can someone please explain what determines whether or not you can have a group homomorphism from one set to another, and then what determines how many you can have?

Answer 1

Are you talking group or ring homomorphisms? It would help to remember that $\mathbb Z/20\mathbb Z$ is generated by $1$ and whatever you map $1$ to should satisfy that adding itself twenty times yields 0. Can you proceed?

Answer 2

Here observe that order of the element should divide both group's order And gcd(20,10)=10 Divisors of 10 are 1,2,5,10 Now try to find those elements in Z10 with these orders and check which of them will give group homomorphism In short the idea is to first find the gcd and ans is 10. No. Of onto homomorphism will be phi(10)=4 No. Of one one homomorphism will be phi(20)=8

Answer 3

The kernel of every homomorphism $\mathbb{Z}\to\mathbb{Z}/10\mathbb{Z}$ contains $10\mathbb{Z}$, hence also $20\mathbb{Z}$.

How many distinct homomorphisms $\mathbb{Z}\to\mathbb{Z}/10\mathbb{Z}$ are there?

What do the homomorphism theorems say?