Lecture 03

Powers and Orders

Definition: Order

Let $G$ be a group.

The order of $G$ is defined as the cardinality of $G$ as a set. Denoted as $|G|$

$|G|<\infty ⟹$ $G$ is a finite group

$|G|=\infty ⟹$ $G$ is an infinite group

e.g. $G={\mathbb{Z}}_5$

$\{0,1,2,3,4\}$

as an additive group $|G|=5$ thus $G$ is finite.

$\mathbb{Z}$ is an infinite group.

Definition: Powers

Let $a\in G,n\in {\mathbb{Z}}_{+}$

$a^n=\underset{n\text{ copies}}{\underbrace{a\cdot a\cdot \cdots \cdot a}}$

$a^0=e$ (the identity)

$a^{-n}=(a^n{)}^{-1}$

e.g. $U(20)=\{1\le x\le 19|\text{gcd}(x,20)=1\}$

Multiplicative

$7^3=7\cdot 7\cdot 7=49\cdot 7$

$=9\cdot 7$

$=63$

$=3$

and

$7^{-3}=(7^3{)}^{-1}=3^{-1}$

$=7$

Theorem: Index Laws

Let $G$ be a group, $m,n\in \mathbb{Z}$

$a^m\cdot a^n=a^{m+n}$

$(a^m{)}^n=a^{mn}$

$(a^{-n})=(a^{-1}{)}^n$

Definition

$G$ is a cyclic group iff:

$\exists a\in G$ s.t. $\forall x\in G$

$x=a^n$ for some $n\in \mathbb{Z}$

we write

$G=⟨a⟩⟺$ $G$ is generated by $a$

e.g. $(\mathbb{Z},+)$

$\mathbb{Z}=⟨1⟩$

Theorem: Every Cyclic Group is Abelian

Let $G$ be a cyclic group, then $G$ is abelian.

Proof

Suppose $x,y\in G$

Then

$x=a^n,$

$y=a^m$

for some $a\in G,m,n\in \mathbb{Z}$

$x\cdot y=a^n\cdot a^m=a^{n+m}=a^{m+n}=a^m\cdot a^n=y\cdot x$

$⟹$ $G$ is Abelian.

e.g.

$U(10)={\mathbb{Z}}_{10}^{\ast }$ is cyclic

$U(8)={\mathbb{Z}}_8^{\ast }$ is not

Definition

Let $G$ be a group and $a\in G$, then

the order of $a$ is the smallest integer $n\in {\mathbb{Z}}_{+}$ s.t. $a^n=e$

denote it $|a|=n$

Remark

if no such $n$ exists for $a$, then $a$ has infinite order.

Theorem

Let $G$ be a group,

and $a\in G$ ,

and $i,j\in \mathbb{Z}$ , then:

$\boxed{1}$ if $a$ has infinite order then

$a^i=a^j⟺i=j$

$\boxed{2}$ if $|a|=n<\infty$, then

$a^i=a^j⟺i\equiv j\mathrm{mod}n$

Proof

$\boxed{1}$ Assume…

$\boxed{2}$

$(⟸)$ trivial

$(⟹)$ Suppose that $i\not\equiv j\mathrm{mod}n$

WLOG: Assume $i>j$

$i-j=n\cdot k+r$

$r<n$