A lot of solutions to problems say that for a cyclic group, such as $\mathbb{Z}/\mathbb{Z}_3$, $\mathbb{Z}/\mathbb{Z}_{10}$, etc., a group homomorphism $\phi$ from $\mathbb{Z}/\mathbb{Z}_m$ to $\mathbb{Z}/\mathbb{Z}_n$ is determined by $\phi(1)$, but I never really understood why... can someone help me? Thanks so much in advance!
Why is group homomorphism determined by $\phi(1)$?
1
$\begingroup$
abstract-algebra
2 Answers
3
Because $\Bbb Z/n\Bbb Z$ is generated by $1$. Thus, $\varphi(2)=\varphi(1+1)=\varphi(1)+\varphi(1)$, and similarly for every other element of $\Bbb Z/n\Bbb Z$.
-
0I just got the notification for this new post, so postponed. But thank you so much! It was really helpful! – 2012-10-11
1
Hint: For example $\varphi(2)=\varphi(1+1)=\varphi(1)+\varphi(1)$