I want to determine the order of each element in the additive group $\mathbb{Z}/9\mathbb{Z}=\lbrace \bar{0},\bar{1},\bar{2},\dots, \bar{8}\rbrace$. It is taken from the Example (4) page 20 in Abstract Algebra. It only has explained the orders of $\bar{5}$ and $\bar{6}$. Though it confuses me to comprehend the Definition (see below) on the same page about $x$ in a group being of order $n\in \mathbb{N}$.
For $G$ a group and $x\in G$ define the order of $x$ to be the smalles positive integer $n$ such that $x^{n}=1$, and denote this integer by $|x|$.
I am fully aware that $\bar{0}$ is an identity of this group, so $|\bar{0}|=1$. After having read the example I thought about finding a $n\in \mathbb{N}$ such that $\bar{k}+\dots +\bar{k}=\overline{nk}=\bar{0}$, where $k\in \mathbb{Z}/9\mathbb{Z}$. We have $|\bar{1}|=9$ because $\bar{1}+\dots+\bar{1}=\overline{9\cdot 1}=\bar{9}=\bar{0}$. Consequently, $|\bar{2}|=|\bar{4}|=|\bar{5}|=|\bar{7}|=|\bar{8}|=9$ and $|\bar{3}|=|\bar{6}|=3$.
The main question is, how do I understand the Definition in compared to this example? When I read the Example (5) it gave me an idea how to understand it a little. If we take the element $\bar{6}$ that has order $3$, does it then mean $6^{3}\equiv 0\pmod{9}$? If I try to do the same to the other orders, some of them fail to satisfy the Definition. For example the element $\bar{1}$ that has order $9$ and it is untrue that $1^{9}\equiv 0\pmod{9}$.
Could you please eleborate how to determine the order if I am wrong and how the Definition should be understood when it comes to this Example?
Thank you.
Best Answer
The problem is that the definition is for multiplicative notation, while you consider an additive group (what makes the situation worse is that multiplication also makes sense for residue classes).
For a group where you use additive notation the defintion transforms to:
The point is you always want to know how often do I have to combined (according to the group law) the element with itself to get the neutral element of the group.
For residue classes you need to be extra careful to know if you consider addition or multiplication. You consider addition. And, so for order of $\overline{6}$ equal $3$ you care about $\overline{6} + \overline{6} + \overline{6} = \overline{0}$, or written differently $3 \ \overline{6} =\overline{0}$.