Definitions
Cyclic group
A group $G$ is cyclic if there exists $a\in G$ such that $\langle a \rangle = G.$
The Order of A
The smallest positive integer $n$ such that $a^n = e$ . If there is no such integer we say $a$ has infinite order, we write $\mid a \mid = n$. Note that: $\mid a \mid = \mid\langle a \rangle\mid$ .
Theorems
Theorem 3.1
Let $a\in G$ with $G$ as a group.
Then $\langle a \rangle:=\{a^k:k\in \Bbb{Z}\}$ is a subgroup of $G$. (or $\{ka:k\in \Bbb{Z}\}$ if operation is addition)
Furthermore, $\langle a \rangle$ is the smallest subgroup containing $a$.
Theorem 3.2
Every cyclic group is abelian.
Theorem 3.3
Every subgroup of a cyclic group is cyclic.
Corollary 3.4
The subgroups of $\Bbb{Z}$ are $n\Bbb{Z}$ where $n=0,1,2, ...$
Proposition 3.5
Let G be a cyclic group of order n and suppose that $a$ is
a generator for G. Then $a^{k} = e$ if and only if n divides k.
Theorem 3.6
Let $G$ be a cyclic group of order n and suppose that $a\in G$
is a generator of the group. If $b = a^k$, then the order of b is $n/d$, where
$d = gcd(k, n)$.
Corollary 3.7
The generators of $\Bbb{Z}_n$ are the integers $r$ such that $1\leq r<n$ and gcd$(r,n)=1$
Examples & Notes
Examples of abelian's and cyclics
$D_3$ has 6 generators, is not abelian and not cyclic
$V_4$ is abelian and not cyclic
$Q_8$ is not abelian and not cyclic
$D_4$ has 8 generators, is not abelian and not cyclic
$\Bbb{Z}$ is abelian and cyclic and is a group of infinite order
$V_4$ the symetry group for a rectangle is cyclic but not abelian. An example of a smallest group from $V_$$ is $<h>$ = $\{{e,h}\}$ where $h$ represents a horizontal flip. The order of $<h>$ or $\mid<h>\mid$ = 2
The converse of theorem 3.2 is false
The statement: every abelian group is cyclic is false counter examples; $V_4$ & $U(8)$
3.37
Prove that if $G$ has no proper nontrivial subgroups, then $G$ is cyclic groups.
Proof: If $g \in G$ where $g \neq e$ & $<g> = G$ is a proper subgroup of $G$. Therefore, $g$ is a generator of $G$ and thus, $G$ is cyclic. $qed$