# What are a few examples of noncyclic finite groups?

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.

• check out the so-called "klein group" of four elements all satisfying $g^2=e$. i'm not sure if the name comes from Felix Klein, or from the German word for "small" – David Holden Dec 11 '13 at 8:36
• Dihedral groups, symmetric groups, alternating groups, groups of matrices over finite fields. – Gerry Myerson Dec 11 '13 at 8:37
• @David, that's a direct product. – Gerry Myerson Dec 11 '13 at 8:37
• Hear, hear! Hear, hear! (Redundancy supplies 15 characters!) – Robert Lewis Dec 11 '13 at 8:38
• you are right, Gerry. as often i "didn't read the question thoroughly". apologies. – David Holden Dec 11 '13 at 8:39

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.

• link to wikipedia page? en.wikipedia.org/wiki/Klein_four-group as their explanation (and the Cayley table really helps!) – Jason S Jun 22 '17 at 14:59
• Is "Klein V group" a typo? (I'm not sure what you were going for if it is a typo though. "Klein four-group $V$"? Although I have only previously seen $V_4$ used, not $V$, until I looked at the wiki article just now. Possibly I have seen $V$ used before and not clocked it though.) – user1729 Jan 25 '19 at 11:41
• The Klein $V$ group is called such because the German term for the number 4 is "vier", – Harry Evans Sep 25 '19 at 4:22

Every finite group is isomorphic to a subgroup of a symmetric group. The symmetric groups themselves are good examples.

• maybe a silly question, but is what you say in fact restricted to finite groups? – David Holden Dec 11 '13 at 8:54
• @DavidHolden, that is Cayley's Theorem and I don't think it is restricted to finite groups. – David Dec 11 '13 at 8:56
• Indeed it's not restricted to finite groups: every group $G$ acts by (e.g.) right multiplication on itself, giving an embedding of $G$ in the symmetric group on $G$. – Shane O Rourke Dec 11 '13 at 8:59
• @DavidHolden How do you define even and odd for permutations of an infinite set? – Carsten S Dec 11 '13 at 9:14
• @DavidHolden: Scott's Group Theory textbook describes these groups very well, including their normal structure. – Jack Schmidt Dec 12 '13 at 14:49