Group Theory - Quotient group notation?

Question

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?

Answer 1

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.

Answer 2

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).