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What is the difference between the following terms:

$\mathbb{Z}_{4}$ , $\mathbb{Z}/4$ and $\mathbb{Z}/{4}\mathbb{Z}$ ?

I am pretty sure the first one is the cyclic group with addition modulo 4... but what about the other two? What does each term mean?

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They are all the same group. It is almost pure notation, but perhaps the most explicit one is $\mathbb{Z}/4\mathbb{Z}$. $4\mathbb{Z}$ is "four times the integers" so $\{-4,0,4,8,12,...\}$ so the cosets are

$$\{...,-4,0,-4,...\}$$ $$\{...,-3,1,5,...\}$$ $$\{...,-2,2,6,...\}$$ $$\{...,-1,3,7,...\}$$

Four cosets and $\mathbb{Z}/4\mathbb{Z}=\{0,1,2,3\}$ under addition. $\mathbb{Z}/4$ is more like "integers divided by four" but the result is the same, and $\mathbb{Z}_4$ is more like "cyclic group of order 4" but again, they are all isomorphic.

EDIT: Details. I get those cosets by picking each element of $\mathbb{Z}$ and adding it to the subset $4\mathbb{Z}$ which I've defined above. So, pick 2 and you get

$$2+4\mathbb{Z}=\{...,-6,-2,0,2,6,10,...\}$$ which is the third coset. $\mathbb{Z}/4\mathbb{Z}$ means "the integers with the cosets identified". So call each coset by a representative member - $\{0,1,2,3\}$ for example (I could also pick $\{-2,-1,0,1\}$ or any other sequence). Under addition this is a group - for example, adding the second and third gets you the forth:

$$1+2=\{...,-3,1,5,9,...\}+\{...,-2,2,6,10,...\}=\{...,-5,-1,3,7,...\}$$ (I'm adding together every element with every other). Hopefully that expression is not too confusing - the left side is the elements $1$ and $2$ in the quotient group $\mathbb{Z}/4\mathbb{Z}$, which I then write in terms of the cosets I've defined above to determine the result of the group action.

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How did you get those cosets? I see you have used 0, 1, 2 and 3, but did you assume addition modulo 4? –  Harold Mar 22 '14 at 15:50
Pick an element of $\mathbb{Z}$ - say 2. Now add each element of $4\mathbb{Z}$ to 2 - you get $\{...,-2,2,6,10,...\}$ so this is the third coset in my list. Do this for every element in $\mathbb{Z}$. Each of them will produce one of the four cosets I've listed. –  levitopher Mar 22 '14 at 15:56
Why is ${\mathbb Z}/4$ the same as ${\mathbb Z}_4$?? Isn't ${\mathbb Z}_4 \simeq \{0,1,2,3\}$ and ${\mathbb Z}/4 \simeq \{ 0, \pm \frac{1}{4}, \pm \frac{1}{2}, \pm \frac{3}{4}, \cdots \}$ ? –  Prahar Mar 22 '14 at 15:57
Thanks. OK so I see how you get those cosets, but I don't see how you jumped to "Four cosets and $\mathbb{Z}/{4}\mathbb{Z}$={0,1,2,3}" ? –  Harold Mar 22 '14 at 16:00
@Prahar: that's why I said it's mostly notation for the same thing, but if you want to think of that as "integers divided by four" you need to subtract off the integer part of each fraction. Then you will get, for instance, $1/4\sim 5/4\sim 9/4...$ etc. –  levitopher Mar 22 '14 at 16:12

The notation $\mathbb{Z}_p$ is used for p-adic integers, while commutative algebraists and algebraic geometers like to use $\mathbb{Z}_{p}$ for the integers localized about a prime ideal $p$ (Fourth bullet point). These are two reasons why use of $\mathbb{Z}_p$ is discouraged for integers mod $p$.

When studying ring theory, ideals of $\mathbb{Z}$ are generated by integers $n$ which are denoted $(n)$, so I think this is the reason for the notations $\mathbb{Z}/(n)$ and $\mathbb{Z}/n$.

Thus $\mathbb{Z}/n\mathbb{Z}$ most commonly refers to the group of integers mod $n$ (i.e. focusing on additive structure), $\mathbb{Z}/(n)$ refers to the ring of integers mod $n$ (i.e. additive and multiplicative structure), and $\mathbb{Z}_n$ can refer to several things (the correct one should be clear from context).

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I would rather use $\Bbb Z_p$ to denote the integers localized at the multiplicative subset $\{1,p,p^2,...\}$. The integers localized at the (complement of the) prime ideal $(p)$ should imho be denoted $\Bbb Z_{(p)}$. –  Nils Matthes Mar 22 '14 at 17:48

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