The importance and applications of order of a group?

Question

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

• If an abelian group has subgroups of orders $m$ and $n$, respectively, then it has a subgroup whose order is $\operatorname{lcm}(m,n)$.

• A simple proof of Sylow theorem for abelian groups

• If a finite group $G$ of order $n$ has at most one subgroup of each order $d|n$, then $G$ is cyclic

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!

Answer 1

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.