Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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 $\Bbb Z/n\Bbb Z$.

share|cite|improve this answer
Thanks, this all makes sense but I guess my question is more what exactly does φ(1) determine and tell us specifically about this specific homomorphism? I am so lost =/ – arcastar Oct 11 '12 at 5:36
@arcastar: $\varphi(1)$ completely determines the homomorphism. This simply means that if you know $\varphi(1)$, then you know $\varphi(a)$ for every $a\in\Bbb Z/n\Bbb Z$; in other words, you know the whole function. That one value pins down every value, leaving no wiggle room. – Brian M. Scott Oct 11 '12 at 5:38
I just got the notification for this new post, so postponed. But thank you so much! It was really helpful! – arcastar Oct 11 '12 at 6:52

Hint: For example $$\varphi(2)=\varphi(1+1)=\varphi(1)+\varphi(1)$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.