6. Homomorphisms and Isomorphisms

Kernels And Images — Quiz

Test your understanding of kernels and images with 5 practice questions.

Practice Questions

Question 1

For a group homomorphism $\varphi:G\to H$, what is the kernel?

Question 2

Let $\varphi:\mathbb{Z}\to\mathbb{Z}_6$ be defined by $\varphi(n)=n\bmod 6$. What is $\ker(\varphi)$?

Question 3

Let $\psi:\mathbb{Z}\to\mathbb{Z}$ be given by $\psi(n)=2n$. What is $\operatorname{im}(\psi)$?

Question 4

If a group homomorphism $\varphi:G\to H$ is injective, what is $\ker(\varphi)$?

Question 5

Which statement about the kernel of a group homomorphism is always true?