3.Order of element

Overview

Group:

$$\begin{gathered} <Z_n,\oplus >,n=6 \\ {Z}_n=\{0,1,2,3,4,5\} \end{gathered}$$

$\oplus$ means plus and mod 6. For example:

identity element is 0. Because adding any number to 0 and taking the remainder of 6 doesn't change anything.

Now let $a=1$, $a^{-3}=(a^{-1}{)}^3=5^3=5\oplus 5\oplus 5mod6=15mod6=3$.

$a^{-1}=5$, because $1\oplus 5mod6=0$​.

a	...	$a^{-6}$	$a^{-5}$	$a^{-4}$	$a^{-3}$	$a^{-2}$	$a^{-1}$	$a^0$	$a^1$	$a^2$	$a^3$	$a^4$	$a^5$	$a^6$	...
0	...	0	0	0	0	0	0	0	0	0	0	0	0	0	...
1	...	0	1	2	3	4	5	0	1	2	3	4	5	0	...
2	...	0	2	4	0	2	4	0	2	4	0	2	4	0	...
3	...	0	3	0	3	0	3	0	3	0	3	0	3	0	...
4	...	0	4	2	0	4	2	0	4	2	0	4	2	0	...
5	...	0	5	4	3	2	1	0	5	4	3	2	1	0	...

It can be seen that the power value of the element changes periodically, and we call this period of change the order of the element.

definition

Let $e$ be the identity element of group $<G,\ast >$, $a\in G$,

(1) The smallest positive integer $n$ which satisfies $a^n=e$ called the order(or tras) of element $a$, denoted as $|a|$​.

(2) If not exist positive integer $n$ that makes $a^n=e$, then the order of $a$​ is infinite.

The order of identity element is always 1.

Example

Find the order of element in group $<\mathbb{R},+>$.

Identity element is 0, $|0|=1$.

For $\mathrm{\forall }a\in \mathbb{R}$, $a\ne 0$, and $\mathrm{\forall }n\in {\mathbb{Z}}^{+}$:

$$a^n=a^{n-1}+a=a+a^{n-1}=a+a+...+a=na\ne 0$$

The order of non-zero elements in the addition group of real numbers is infinite.

Property

(1) $\mathrm{\forall }a\in G$, $|a|=m$, then $a^n=e$ if and only if $m|n$ ($n$ can be divided by $m$​).

This property essentially means that powers of $a$ repeat themselves every once in a while.

(2) If group $G$ is finite, then the order of all elements are finite, and can't bigger than order of group $G$.

(3) $\mathrm{\forall }a\in G$, $|a|=|a^{-1}|$.

(4) $\mathrm{\forall }a,b\in G,|a|=m,|b|=n$, if $(m,n)=1,ab=ba$, then $|ab|=mn$.