3
$\begingroup$

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?

$\endgroup$
  • 2
    $\begingroup$ 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. $\endgroup$ – Nap D. Lover Oct 14 '17 at 15:29
15
$\begingroup$

$\mathbb{Z}/6\mathbb{Z}$ 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.

$\endgroup$
2
$\begingroup$

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 $4+4=2$.

$\endgroup$
  • $\begingroup$ Would it not be better to destinguish the elements of $\mathbb{Z}/6\mathbb{Z}$ from integers, to stop confusion? $\endgroup$ – Cornman Oct 14 '17 at 14:13
  • $\begingroup$ 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 $\endgroup$ – Prince Thomas Oct 14 '17 at 14:19
  • 1
    $\begingroup$ @Cornman In this specific case, I don't think that there is a large risk of confusion. $\endgroup$ – José Carlos Santos Oct 14 '17 at 14:19
  • 1
    $\begingroup$ Better safe than sorry. :) $\endgroup$ – Cornman Oct 14 '17 at 14:23
1
$\begingroup$

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

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

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.