Prove that all cyclic groups are Abelian.
For some group $G$, define a cyclic subgroup of $G$ as $\left \langle a \right \rangle = \{a^n : n \in \mathbb{Z}, a \in G\}$
Then it follows that $a^n,a^m \in \left \langle a \right \rangle$ is always true. We wish to show that $\left \langle a \right \rangle$ is abelian, or that $$a^na^m = a^ma^n$$ So... $$a^na^m=a^{n+m}=a^{m+n}=a^ma^n$$ Hence, every cyclic group is Abelian since for all $m,n \in \mathbb{Z}$, $a^na^m=a^ma^n$.
Is that really all we have to do? This proof seems way too easy but also right.
Thanks!