6. Homomorphisms and Isomorphisms

First Isomorphism Theorem — Quiz

Test your understanding of first isomorphism theorem with 5 practice questions.

Practice Questions

Question 1

Which statement is the First Isomorphism Theorem for a group homomorphism $f:G\to H$?

Question 2

If $f:G\to H$ is a homomorphism and $\ker f=\{e\}$, what can be concluded?

Question 3

Let $f:\mathbb Z\to\mathbb Z_n$ be defined by $f(k)=\overline{k}$. What is $\ker f$?

Question 4

For the same map $f:\mathbb Z\to\mathbb Z_n$ with $f(k)=\overline{k}$, what is $\operatorname{im}f$?

Question 5

Let $f:\mathbb Z\to\mathbb Z$ be defined by $f(k)=2k$. What is $\ker f$?