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

  • $\mathbb{Z}/n\mathbb{Z}$ (which denotes a quotient group) 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 GF(n)?

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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
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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
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3 Answers 3

up vote 10 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.

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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
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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
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$C_n$ 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
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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
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while $\mathbb{Z}_{(p)}$ denotes localized at $(p)$... –  wxu Jun 7 '11 at 6:32
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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.

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@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." –  Sadeq 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
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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!).

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