The importance and applications of order of a group?

Recently, I'm exposed to some exercises and theorems concerning order of a group. For example,

IMHO, the classic result of this kind is Sylow theorems that appear in most standard textbooks about abstract algebra. As such, I would like to ask about the importance of order of a group in abstract algebra and its applications in other branches of mathemactics.

Thank you for your elaboration!

• One important application that comes to mind is that the order of the Galois group of a field extension is equal to the degree of the field extension. – Vercassivelaunos Jul 23 at 8:07
• You will find some answers in this nice little document : math.mit.edu/~jwellens/Group%20Theory%20Forum.pdf – Jean Marie Jul 23 at 8:25

The classification of finite simple groups was one of the great mathematical achievements of the 20th Century. It is also one where a single result on the order of the groups played a key role, namely the Feit–Thompson theorem, or odd order theorem:

Theorem. (Feit-Thompson, 1963) Every group of odd order is soluble*.

The proof is famously long, at 255 pages, and has recently been Coq-verified [1].

The derived subgroup of a soluble group is a proper normal subgroup, and so a soluble group is simple only if it is abelian. Therefore, the Feit-Thompson theorem has the following corollary:

Corollary. Every non-cyclic finite simple group has even order.

There are other results in this vein, with much shorter proofs. For example, Burnside's theorem (Wikipedia contains a proof):

Theorem. (Burnside, 1904) Let $$p, q, a, b\in\mathbb{N}$$ with $$p, q$$ primes. Then every group of order $$p^aq^b$$ is soluble.

Therefore, every non-cyclic finite simple group must have order divisible by three primes. Moreover, at least one of these primes occurs twice in the prime decomposition of the order:

Theorem. (Frobenius, 1893) Groups of square-free order are soluble.

You can find a proof of this theorem on Math.SE here. The answer there links to the article [2], where the theorem is Proposition 17 (page 9). The article also claims that the result is due to Frobenius in [3].

*In American English, solvable.

[1] Gonthier, Georges, et al. "A machine-checked proof of the odd order theorem." International Conference on Interactive Theorem Proving. Springer, Berlin, Heidelberg, 2013.

[2] Ganev, Iordan. "Groups of a Square-Free Order." Rose-Hulman Undergraduate Mathematics Journal 11.1 (2010): 7 (link)

[3] Frobenius, F. G. "Uber auflösbare Gruppen." Sitzungsberichte der Akademie der Wiss. zu Berlin (1893): 337-345.

• If I remember correctly the origin of the square-free order theorem is before Burnside, but it certainly follows from the following result of his, from around 1900: if $p$ is the smallest prime dividing a simple group $G$, then $p^3\mid |G|$ or $p=2$ and $12\mid |G|$. This was the origin for Burnside's conjectures that 1) $2\mid |G|$ for simple $G$ (true) and $3\mid |G|$ for simple $G$ (false). This result follows from Burnside's normal $p$-complement theorem. – David A. Craven Jul 23 at 13:21
• @DavidCraven I dug a bit deeper into the answer I linked too (which I probably should have done before!). I've now added in an article which claims that the result is due to Frobenius, although his paper is in German and I could not find an online version, so I have not verified this claim. The paper is: Frobenius, F. G. "Uber auflösbare Gruppen." Sitzungsberichte der Akademie der Wiss. zu Berlin (1893): 337-345. – user1729 Jul 23 at 13:26
• I don't have that paper of Frobenius's myself. Howerver, I do have Burnside's. In a paper of his from 1895 he also proves the result, and states that 'Herr Frobenius' proved it in that paper in May 1893. I cannot entirely trust that paper though, since apparently it was received on February 18th, and read on February 14th... – David A. Craven Jul 23 at 13:33
• @David That reminds me of a story I heard of a PhD student who submitted to a middling journal. The editor was in the same institution, so walked across the hallway to the student's supervisor's office to ask if the paper was correct. The supervisor assured them that it was correct, so the paper was accepted there and then. The PhD student is now a professor, and has never submitted to this journal since. – user1729 Jul 23 at 13:40
• (Actually, that last sentence is not true. My memory of the story ends like this, but I just checked the professor-in-question's website and they published a second paper in that journal a few years later. Pity.) – user1729 Jul 23 at 13:44