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There are many results, but whether you would find them interesting is up to you. (I don't even know what it would mean for a result to not be a "statement" or

There are many results, but whether you would find them interesting is up to you. (I don't even know what it would mean for a result to not be a "statement" or "observation". Every group-theoretic result is a statement of an observation, and for that matter, the Riemann Hypothesis is a statement too.) Since you mention Lagrange's theorem, I presume you are looking for introductory results rather than advanced results. E.g., the result that a functor has at most one left adjoint up to isomorphism (a direct generalization of the fact that inverses are unique in groups). Or the result that a functor which has a left adjoint automatically preserves limits (and the near converse to this given by the Adjoint Functor Theorem). Or the theorem that a category has a limit for the entire category if and only if it has an initial object. Or the Yoneda lemma, which is a direct generalization of Cayley's theorem for groups. Or Lambek's theorem that every initial algebra is invertible. What is a group-theoretic result of interest to you, apart from the aforementioned Lagrange's theorem? It might in particular be good to see a group-theoretic result which you find interesting even outside the context of finite groups (as there is not such an emphasis on considering finiteness in typical category theory, though one could of course say things specifically about finite categories too).

Chinjut 2y ago
In particular, you seem fond of the application of Lagrange's theorem to deriving facts about prime numbers. Presumably, what you have in mind here is the proof of Fermat's little theorem by saying "The integers modulo prime p under multiplication form a monoid of size p, among which the invertible subgroup comprises precisely the values not equivalent to zero (thus, of size p - 1). Within these, the order e of any value x is equivalent to the size of the subgroup x generates, which by Lagrange's theorem (decomposing the entire group into its cosets; i.e., its isomorphic orbits under the subgroup acting freely upon it) must divide the size of the overall group. Thus, x^(p - 1) = 1 mod p". Yes, you can say all that in group-theoretic language. But someone skeptical of the value of group theory might object "All you are doing is using fancy words to say what could be said more straightforwardly. After all, Fermat's little theorem was understood by Fermat and others two hundred years before group theory was invented. Euler published his generalization of Fermat's little theorem a hundred years before group theory was invented. All your fancy-schmancy group theory is just pointlessly obfuscatory language for describing something already well understood". And indeed, all group theory is doing in this case is showing us how to relate this one particular result to many other similar results. But that is the value of abstraction and generalization. That is how math often works. Recognizing a common pattern that comes up repeatedly, making particular instances obvious in retrospect, through fluently understanding their shared connection, once and for all. That's how it worked for groups, and that's the same way it works for categories.

Chinjut 2y ago

In particular, you seem fond of the application of Lagrange's theorem to deriving facts about prime numbers. Presumably, what you have in mind here is the proof of Fermat's little theorem by saying "The integers modulo prime p under multiplication form a monoid of size p, among which the invertible subgroup comprises precisely the values not equivalent to zero (thus, of size p - 1). Within these, the order e of any value x is equivalent to the size of the subgroup x generates, which by Lagrange's theorem (decomposing the entire group into its cosets; i.e., its isomorphic orbits under the subgroup acting freely upon it) must divide the size of the overall group. Thus, x^(p - 1) = 1 mod p". Yes, you can say all that in group-theoretic language. But someone skeptical of the value of group theory might object "All you are doing is using fancy words to say what could be said more straightforwardly. After all, Fermat's little theorem was understood by Fermat and others two hundred years before group theory was invented. Euler published his generalization of Fermat's little theorem a hundred years before group theory was invented. All your fancy-schmancy group theory is just pointlessly obfuscatory language for describing something already well understood". And indeed, all group theory is doing in this case is showing us how to relate this one particular result to many other similar results. But that is the value of abstraction and generalization. That is how math often works. Recognizing a common pattern that comes up repeatedly, making particular instances obvious in retrospect, through fluently understanding their shared connection, once and for all. That's how it worked for groups, and that's the same way it works for categories.