What is the meaning of Z/2017Z in abstract algebra?

Question

I am trying to understand what

$\mathbb{Z}/2017\mathbb{Z}$ means.

Is it related to mod 2017 in some way?

I am not sure if / symbol here means division so I am a bit confused.

$\mathbb{Z}$ = {...,-2,-1,0,1,2,...}

2017$\mathbb{Z}$ = {...,-4034,-2017,0,2017,4034,...}

$\mathbb{Z}/2017\mathbb{Z}$ = ??

Could anybody help me?

Answer 1

$\mathbb{Z}/n\mathbb{Z}$ means the ring of integers modulo $n$. It is the set of all congruence classes of the integers modulo $n$. See Wikipedia.

When $n$ is prime, which is the case for $2017$, then $\mathbb{Z}/n\mathbb{Z}$ is a field.

Answer 2

This is an example of a quotient group. You may have already heard about modular arithmetic; well, expressions like "$G/H$" describe quotients of groups, which is a process analogous to how we get from the integers to the integers modulo [something].

Specifically, $\mathbb{Z}/2017\mathbb{Z}$ is just the integers modulo $2017$. Here's how the syntax works, informally:

• We have two "number systems" $G$ and $H$; these should be things where we have a notion of addition or something similar. For example, above $G$ is $\mathbb{Z}$ and $H$ is $2017\mathbb{Z}$ (which itself is just a notation for the set of integer multiples of $2017$, as you guessed).

• We also need $H$ to be contained in $G$. Actually, we need a bit more than this - we need that $H$ sits inside $G$ in a "nice" way - but that gets a bit technical, and is irrelevant as long as addition (in whatever sense we're using it) is commutative.

• "$G/H$" then denotes "$G$ modulo $H$"; we think of two elements $g_1,g_2$ of $G$ as being equivalent if their difference is in $H$. The analogy with modular arithmetic is that two integers $a, b$ are equal modulo $p$ if their difference is a multiple of $p$ (so for "addition mod $p$," our $H$ should just be $p\mathbb{Z}$).

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To keep things simple, I've described above the quotient of groups. However, there is lots of structure in the integers besides addition - in particular, multiplication! Just like addition, multiplication also makes sense "modulo $p$," and if we take this additional structure into account we need to talk about quotients of rings instead of just groups.

Answer 3

It's related to $\mod 2017$ in every way.

It is the group of equivalence classes $\mod 2017$

It is $\{[0],[1],[2], ...... ,[2016]\}$ where each of the $[i] := \{..., i - 2*2017, i - 2017, i , i + 2017, i + 2*2017,....\}=\{z\in \mathbb Z| z \equiv i \mod 2017\}$.

Answer 4

In general, $ \mathbb{Z} / n \mathbb{Z} $ is the set of equivalence classes of $ \mathbb Z $ with respect to the relation $ a \sim b \iff n | a - b $, so the relation is equivalence modulo $ n $. We also endow it with an additive and multiplicative structure via $ \overline{a} + \overline{b} := \overline{a + b} $ and $ \overline{a} \overline{b} := \overline{a b} $, where $ \overline{a} $ is the equivalence class of $ a $. One can check that these operations are well defined.