After defining cyclic groups and generators, the next structural result is that cyclic groups are always abelian. This theorem is powerful because it turns the existence of a single generator into a commutativity law for the whole group. The proof is short but conceptually important: if every element is a power of the same element, then any two elements commute because integer addition in the exponents is commutative. In this lesson, students will prove that every cyclic group is abelian and learn why the converse is false.

Let $(G,\circ)$ be a group. If $G$ is cyclic, then $G$ is abelian.