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

I have difficulties understanding the difference between the following two notations:

  • $\mathbb{Z}/n\mathbb{Z}$ (which denotes a quotient ring) and
  • $\mathbb{Z}_n$.

Are they equivalent?

PS1: The same applies to the multiplicative counterparts:

  • $(\mathbb{Z}/n\mathbb{Z})^*$
  • $\mathbb{Z}_n^*$.

PS2: It can be proven that $\mathbb{Z}/n\mathbb{Z}$ is a field if and only if $n$ is prime. Assuming $n$ is prime, could you compare $\mathbb{Z}/n\mathbb{Z}$ with $\text{GF}(n)$?

share|cite|improve this question
Yes, they are. But $\mathbb{Z}_p$ often denoted $p$-adic integers which is not the same as $\mathbb{Z}/p\mathbb{Z}$ at all. Some author(e.g. Rotman) also uses $\mathbb{I}_n$ denote $\mathbb{Z}/n\mathbb{Z}$ – wxu Jun 7 '11 at 6:20
Usually I use the notation $GF(p)$ or $\mathbb F_p$ when I'm explicitly interested in the field structure. For the additive group structure I prefer $\mathbb Z/p\mathbb Z$ or $C_p$. – Giacomo d'Antonio Jun 7 '11 at 8:02
up vote 11 down vote accepted

It depends on the textbook/paper author, but often $\mathbf{Z}/n\mathbf{Z}$ and $\mathbf{Z}_n$ mean the same thing.

A word of caution, however: using the notation $\mathbf{Z}_n$ to mean $\mathbf{Z}/n\mathbf{Z}$ can cause confusion, because $\mathbf{Z}_p$ is also used to denote the p-adic integers. Thus, many mathematicians (especially number theorists) reserve the shorter notation for p-adics and use the long notation for the finite cyclic groups.

Edit: Just now saw your second question. The answer is that, indeed, $\mathbf{Z}/p\mathbf{Z} = GF(p)$, where $p$ is prime.

share|cite|improve this answer
For some reason, some authors and teachers think that they will scare first year undergraduates by using the quotient group/ring notation. That is the notation one should exclusively though, for the reason you describe. – Alex B. Jun 7 '11 at 6:17
I agree! Another notation which I don't often see, but which I sort of favor, is $\mathbf{C}_n$ for the cyclic group of order n. It has the same brevity as $\mathbf{Z}_n$ without the confusion. Unless $\mathbf{C}_n$ is used for something else, too...? – Jeff Jun 7 '11 at 6:20
$C_n$ or $Z_n$ (for German zyklisch) is fine and is often used, but $\mathbb{C}_p$ is used in number theory for the completion of the algebraic closure of the field of $p$-adic numbers, so shouldn't be used in this context. – Alex B. Jun 7 '11 at 6:22
Good points, all. As long as authors are explicit at the offset about what notation they are using, I guess it needn't cause any confusion. As always, context is key. – Jeff Jun 7 '11 at 6:25
while $\mathbb{Z}_{(p)}$ denotes localized at $(p)$... – wxu Jun 7 '11 at 6:32

If $n$ is a prime number, then $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$ are isomorphic (in fact I would simply define $GF(n)=\mathbb{Z}/n\mathbb{Z}$ when $n$ is a prime number).

However, if $n$ is some power of a prime number, say $n=p^k$ for $k\geq 2$, then $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$ are not the same.

share|cite|improve this answer
@Zev using the undefined notation $GF(n)$ that didn't feature anywhere in the question and not answering the actual question is likely to exacerbate the OP's confusion instead of clearing it up. – Alex B. Jun 7 '11 at 6:19
@Alex: The OP amended their question to also ask for a comparison of $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$. – Zev Chonoles Jun 7 '11 at 6:21
@Zev: sorry Zev, I hadn't see the edit. +1 then :-) – Alex B. Jun 7 '11 at 6:24
Thanks. As you pointed, the wording "it is said" is not good. I changed it to "it can be proven." – M.S. Dousti Jun 7 '11 at 6:27
@Sadeq: I've removed the corresponding part of my answer, now that it is changed. – Zev Chonoles Jun 7 '11 at 6:27

The notations are equivalent if the author has been careful enough to tell you that by $Z_n$ she means "the integers modulo $n$." If she has not been careful than you have to study the context to decide whether the author means the integers modulo $n$ or something else.

By the way, $Z/nZ$ is not just a quotient group, it's a quotient $\it ring$ (if you haven't studied rings and ideals yet, you have something to look forward to!).

share|cite|improve this answer

To avoid confusion that mentioned in Jeff's answer, some contemporary textbooks (like Rotman's Advanced Modern Algebra) use $\mathbb I_n$ instead of $\mathbb Z_n$. The symbol $\mathbb I$ is the first letter of integer.

share|cite|improve this answer
This was mentioned in a comment to the original question. Five years ago, ten minutes after the question was posted. – tomasz Mar 24 at 8:09
@tomasz :) I only read the question and the answers! – user217174 Mar 24 at 8:11

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.