I just want to make sure that every group $\mathbb{Z}_n$ for any $n$ is cyclic. Further, every group of prime order is cyclic because it is isomorphic to $\mathbb{Z}_p$. I think I have a handle on that, but what other finite groups are there that aren't cyclic? In particular, I'm looking for examples that aren't direct products of groups, just simply groups like $\mathbb{Z}_n$, not $\mathbb{Z}_a \times \mathbb{Z}_b$, etc.
Abstract Algebra – Examples of Noncyclic Finite Groups
abstract-algebrafinite-groupsgroup-theory
Best Answer
The easiest examples are abelian groups, which are direct products of cyclic groups. The Klein V group is the easiest example. It has order $4$ and is isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$. As it turns out, there is a good description of finite abelian groups which totally classifies them by looking at the prime factorization of their orders.
The very first nonabelian group you run into will usually be dihedral groups, the symmetries of $n$-gons. These are just cyclic groups that can be "flipped" by a certain element called a reflection.
Next step up is symmetric groups, the group of bijective functions from $\{1,\ldots,n\}$ to itself (also known as permutations). Cayley's theorem shows how these relate to finite groups in general. (This is not a very useful result, though, for practical purposes, as permutation representations of groups are often the worst possible way to understand the group's structure.) Alternating groups are related to symmetric groups, too, but they're a bit harder to understand from a structural standpoint.
And if you really want to see a lot of examples of groups, you can read my answer here, though a lot of it may be out of your reach at this point.