Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How to prove that the quotient group $\langle \mathbb{C},+\rangle/\langle \mathbb{Z},+\rangle$ isomorphic to the group $\langle \mathbb{C}^*,\cdot\rangle$ based on the first isomorphism theorem?

share|improve this question
I'm not sure that everyone will know what you mean by $C$ here. –  Dylan Moreland Dec 10 '11 at 6:23
Is this a homework question? If so, please add the tag [homework], and preferably tell us what you've tried on your own. –  Asaf Karagila Dec 10 '11 at 11:01

2 Answers 2

Think about the complex exponential map $z \mapsto e^z$. It does much of what you need. From \[ e^{x + iy} = e^x(\cos y + i\sin y), \] can you determine the kernel? It isn't $\mathbf Z$, but perhaps you can scale $z$ first in order to make things work.

share|improve this answer

Just to add a few words to Dylan's excellent answer: why would you think about complex exponentiation?

If you are trying to prove an isomorphism using the First Isomorphism Theorem, then you are trying to define a group homomophism $$f\colon \mathbb{C}\to\mathbb{C}^*$$ that sends sums to products. That is, you are looking for a function $f$ that will satisfy $$f(a+b) = f(a)f(b).$$ This functional equation should immediately suggest the standard functions that have this property: exponential functions. Once you realize that you are looking for an exponential function, it's just a matter of figuring out which exponential function you are looking for.

share|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.