Cyclic Groups

Let $G$ be a group and let $g\in G$. The cyclic subgroup generated by $g$ is
$⟨g⟩=\{g^m:m\in \mathbb{Z}\}$

Lemma: Suppose $g$ has order $n$. Then $⟨g⟩=\{1,g,g^2,...,g^{n-1}\}$
Moreover, $1,g,...,g^{n-1}$ are pairwise distinct, hence $\#⟨g⟩=n$

We say that $G$ is cyclic if $G=⟨g⟩$ for some $g\in G$, which is called the generator of $G$

Lemma: Cyclic groups are Abelian
Proof: Suppose $G=⟨g⟩$. Then

• $g^r\cdot g^s=g^{r+s}=g^{s+r}=g^s\cdot g^r$
  • As integer addition is commutative