# Additive group of integers modulo $6$

I am currently studying Galois Theory and am having trouble understanding group notation.

What does $$\mathbb{Z}/n\mathbb{Z}$$ mean? I understand that its an additive group of modulo $n$ but what would the elements of $$\mathbb{Z}/6\mathbb{Z}$$ be for example?

• You should really consider the italicized paragraph in @lisyarus 's answer. The most common dilemma i see people face is not adequately grasping pre-requisites to newer topics. – Nap D. Lover Oct 14 '17 at 15:29

$\mathbb{Z}/6\mathbb{Z}$ is a quotient group of $\mathbb{Z}$ by the (trivially normal) subgroup $6\mathbb{Z}$, and it's formal elements are 6 cosets: $\{ 0 + 6\mathbb{Z}, 1 + 6\mathbb{Z}, \dots, 5 + 6\mathbb{Z} \}$. These can be identified by chosing a single representative, like $0$ for $0 + 6\mathbb{Z}$ (sometimes denoted $\bar 0$ or $[0]$) and so on.

If you have trouble understanding this, I suggest you to study general group & ring theory first, before trying Galois theory, which is a bit more advanced.

One way of seing this is to say that$$\mathbb{Z}/6\mathbb{Z}=\{0,1,2,3,4,5\}$$and that, if $a,b\in\mathbb{Z}/6\mathbb{Z}$, then $a+b$ is the remainder of the division by $6$ of the usual sum of $a$ and $b$. For instance, $2+3=5$ and $4+4=2$.

• Would it not be better to destinguish the elements of $\mathbb{Z}/6\mathbb{Z}$ from integers, to stop confusion? – Cornman Oct 14 '17 at 14:13
• Its understood when you are talking about modulo groups but you can use the notations like $[0]$ this too in some books bar "-" on the integers is also used to distinguish them from integers – Prince Thomas Oct 14 '17 at 14:19
• @Cornman In this specific case, I don't think that there is a large risk of confusion. – José Carlos Santos Oct 14 '17 at 14:19
• Better safe than sorry. :) – Cornman Oct 14 '17 at 14:23

It is $\mathbb{Z}/6\mathbb{Z}=\{[0],[1],[2],[3],[4],[5]\}$. The residue classes modulo 6

You can add them like this: $[a]+[b]=[a+b]$

The elements are the congruence classes modulo n. For example the elements of the additive group $Z_6$ are $0,1,2,3,4,5$, where $0$=${...-12,-6,0,6,12,...}$ is the set of all integers congruent (mod 6) to $0$ and so on.

The elements of $\mathbb{Z} / 6 \mathbb{Z}$ are often best expressed as integers — the point of the group is that we work modulo the congruence relation where we say that $m \equiv n$ if and only if $m-n \in 6\mathbb{Z}$.

The numbers $0, 1, 2, 3, 4, 5$ cover the entire group, in the sense that every element of $\mathbb{Z} / 6 \mathbb{Z}$ is equivalent to one of those six. Other, equivalent collections such as $12, -5, 20, 3, -2, -25$ also have this property.

That said, when people give rigorous definitions they tend to prefer alternte expressions for things like $\mathbb{Z} / 6 \mathbb{Z}$ for which elements are equivalent if and only if they are actually equal.

There are a number of different strategies for achieving this, which is why you will see lots of conflicting definitions. The two most common strategies, as indicated in the other answers are:

• Use equivalence classes to represent individual elements. E.g. the class $\{ x \in \mathbb{Z} \mid x \equiv 1 \}$ is used to represent the element I referred to above as $1$ (and as $-5$). Usually some form of abbreviated notation is used to denote this class, such as $[1]$ or $\bar{1}$. Then, you have $[1] = [-5]$, so this achieves the goal of reducing equivalence to equality.
• Select a set of "representatives". E.g. use the integers $0,1,2,3,4,5$ to represent elements and never any other integers.