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There is a way to define the notion of prime number in the framework of group theory, thanks to the following observation:
Observation: A non-trivial group $G$ is cyclic of prime order iff for any subgroup $K \le G$ then $K=1$ or $G$.

Now the notion of group and subgroup can be internalized to any category (see group object and subobject). Then, using the above observation, we can internalize the notion of prime number to any category, and call it prime number object.

More precisely, let $C$ be a category with binary products and a terminal object $*$. Let $\operatorname{Grp}(C)$ be the category of group objects in $C$. An object $G$ in $\operatorname{Grp}(C)$ is called a prime number object in $C$ if for any monomorphism in $\operatorname{Grp}(C)$, $i: H \hookrightarrow G$, then $H=*$ or $i$ is an isomorphism.

Question: Is a prime number object always given by the data of a prime number (or a set of prime numbers)?

Remark 1: My guess is yes, but if we relax the notion of group object with the notion of Hopf object (as suggested in Adrien's answer to Is a Hopf algebra a group object of some category?) then I expect the question to become open and hard (see Is there a non-trivial Hopf algebra without left coideal subalgebra?).

Remark 2: We can also define the notion of natural number object (eventually relaxed as above) using the fact that a group is finite iff its subgroup lattice is finite, and Ore's theorem stating that a finite group is cyclic iff its subgroup lattice is distributive.

LSpice
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Sebastien Palcoux
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    Can you please make precise your definition of a prime number object? Also, what do you assume about your category? (I have a guess, but with that definition your question is not well-defined.) – Martin Brandenburg May 31 '21 at 10:18
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    As @MartinBrandenburg says, this depends on your definitions. A simple example: consider the category of sets-with-an-involution-$\tau$. Then prime number objects will be groups $G$ with an involutive group automorphism $\tau:G\to G$ such that for any $g\in G$ with $g\ne1$, $G$ is generated by $g$ and by $\tau g$. On the other hand you have in this category the notion of object of subgroup objects. This is the involutive set of all (not necessarily involutive) subgroups, so if you impose condition that this object reduces to ${1,G}$, then you indeed get only "ordinary" primes. – მამუკა ჯიბლაძე May 31 '21 at 11:26
  • No in fact in the last case each prime $p$ except $2$ will be represented twice: one with trivial involution of $\mathbb Z/p\mathbb Z$ and another with $\tau(x)=-x$. – მამუკა ჯიბლაძე May 31 '21 at 11:35
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    @MartinBrandenburg: Done! Is it ok for you? – Sebastien Palcoux May 31 '21 at 15:06
  • The definition is still not correct, namely 1) I guess you want a prime number object to be an object of $C$ (the underlying object of a group object, I guess), not of $\mathrm{Grp}(C)$, 2) your question if a prime number object is a set of prime numbers is not well-defined (how can an object of an abstract category be a set of prime numbers?). – Martin Brandenburg May 31 '21 at 16:53
  • @MartinBrandenburg: $Grp(C)$ is the category of group objects in $C$. Next, « given by the data of » is different from « is ». – Sebastien Palcoux May 31 '21 at 18:48

1 Answers1

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The question is very vague as it never says what it means by "given by the data of a prime number". Though I think the following example should convince anyone that classyfing such "prime number object" can be much more complicated than listing prime numbers.

Take $K$ to be a fixed group and $\mathcal{C}$ to be the category of $K$-sets (i.e. sets with an action of $K$).

A group object of $\mathcal{C}$ is a group with an action of $K$, and any irreducible representation of $K$ on a $\mathbb{Z}/p\mathbb{Z}$-vector space give you such a "prime number object".

In particular such objects contain much more information than just a prime numbers: their numbers of elements is in general a prime power (or infinite) and a group $K$ can have many different irreducible representations in characteristic $p$, even of the same cardinality.

Note that there might be other kind of "prime number object" in that category that are not even "attached" to a prime number in any reasonable sense: if $K = \mathbb{Q}^*$ is the multiplicative group of rational, then $\mathbb{Q}$ with its natural action of $\mathbb{Q}^*$ and its additive group structure is a "prime number object" in $K$-sets. And it is not related to any prime number.

Simon Henry
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  • If I am not mistaken, in your last example, $(\mathbb{Q}^, \mathbb{Q})$ can be replaced by $(\mathbb{K}^,\mathbb{K})$, with $\mathbb{K}$ any field, or even any division ring. If I am not mistaken, a ring is a division ring iff it is a left simple ring; so I don't think that we can generalize further. – Sebastien Palcoux Jun 03 '21 at 10:09