First, students will learn what cyclic groups are, and see some examples and nonexamples. Here's an excellent video. Then they will see a proof, such as this one, that cyclic groups really are groups. Following that, students will see that abelian does not imply cyclic, but cyclic does imply abelian. Here's a fantastic video on this topic. After that, students will practice proving some things are, or are not, cyclic groups. Here's an example where students are asked to prove a group is cyclic: Let \(\mathbb{Z}_m\) be the set of integers modulo \(m,\) and let \(+_m\) be addition modulo \(m.\) Is \((\mathbb{Z}_m, +_m)\) a group? If it is a group, is it cyclic? The answer can be found here. Here's an easy exercise: Prove \((\mathbb{R}, +\) is not cyclic (video.) Here's a slightly harder exercise: Prove \(\mathbb{Z} \times \mathbb{Z}\) is not cyclic (video). Finally, here are some additional ideas for exercises.
