The task is to find all the Homomorphisms from group $\mathbb{Z}_{20}$ to $\mathbb{Z}_{16}$. My teacher told be that there is an sufficient way to do this, however I've just brute forced it. Do you have any ideas? Would be grateful for the answer also, just to check my own one.
Asked
Active
Viewed 62 times
0
-
2This has been asked very often here (search a bit). For general $n$ and $m$ see this post. – Dietrich Burde Apr 23 '20 at 14:24
-
@DietrichBurde, thanks! And do you know any website that can help me to check my answer? – Marley Corogett Apr 23 '20 at 14:26
-
1Yes, the duplicate. Take $n=20$ and $m=16$. – Dietrich Burde Apr 23 '20 at 14:35
1 Answers
1
Several ideas that might be easier than brute force.
Since the domain is a cyclic group, every homomorphism is determined by where it sends a generator. What are the possibilities?
The quotient of the domain modulo the kernel of a homomorphism will be isomorphic to a subgroup of the codomain.
Ethan Bolker
- 95,224
- 7
- 108
- 199