Section 6 Cyclic Groups

Order

NOTE

$|G|$ is called the order of $G$ For $a\in G$, $ord(a)=min(\{n\ge 1|a^n=e\}\cup \{\infty \})$

Example

${\mathbb{Z}}_p$ has an order of $p$ if $p$ is a prime $|{\mathbb{Z}}_7|=7$

• $ord(1)=7$
• $ord(2)=7$

$|{\mathbb{Z}}_7^{\ast }|=6$

• $ord(1)=1$
• $ord(2)=3$
• $ord(3)=6$

$|\mathbb{Z}|=\infty$

Definition: Cyclic Groups

$⟨a⟩\stackrel{def}{=}\{a^n|n\in \mathbb{Z}\}$ A group $G=⟨g⟩$ generated by $g\in G$ is called a cyclic group

$|{\mathbb{Z}}_7^{\ast }|=⟨3⟩=\{3,2,6,4,5,1\}$

Lemma 5.6

Lemma 5.6

In a finite group G, every element has a finite order.

6.1 Theorem

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=64&selection=318,0,318,30&color=note|p.59]]

Every cyclic group is abelian.

6.6 Theorem

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=66&selection=53,1,54,38&color=note|p.61]]

A subgroup of a cyclic group is cyclic

6.14 Theorem

NOTE

Let G be a cyclic group with n elements and generated by a. Let $b\in G$ and let $b=a^s$. Then b generates a cyclic subgroup H of G containing n/d elements, where d is the greatest common divisor of n and s. Also, $⟨a^s⟩=⟨a^t⟩$ if and only if $gcd(s,n)=gcd(t,n)$

• Proof of 6.14 🔽