No, not every finite p-group can be realized as the group of rational points over 𝔽_p of a connected unipotent algebraic group defined over 𝔽_p. A counterexample is the quaternion group Q₈, a non-abelian 2-group.

Connected unipotent algebraic groups over 𝔽_p are smooth, affine, and their 𝔽_p-rational points form a finite p-group. The order of such a group's rational points is pᵈ, where d is ...

No.

Over a finite field k = Fp, every connected unipotent k-group U is k‑split. In particular, one can choose a central series by connected, closed, normal k‑subgroups U = U1 ▹ U2 ▹ ⋯ ▹ Uc = 1 with Ui/Ui+1 ≅ Ga and with the commutator map satisfying [Ui, Uj] ⊆ Ui+j. For such a “weighted” (or N‑graded) filtration there is a k‑defined cocharacter λ: Gm → Aut(U) that acts on the successive quotients...

As far as we know to-day the question is still open.
No one has yet produced a finite p-group which can be proved not to occur as the group of Fp–rational points of a connected unipotent algebraic Fp–group, but neither has anybody shown that every p-group does occur. What is known can be summarised as follows.

1. All abelian p-groups occur.
   Choose, for every non–negative integer m, ...

Here is what is known, and why the general question remains open.

1. Reformulation in terms of restricted Lie algebras.
Over a perfect field of characteristic p there is an equivalence of categories (Cartier–Kostant–Milnor–Moore, or cf. Waterhouse, Chap. IV) between
a) finite‐dimensional nilpotent restricted Lie algebras
and
b) connected unipotent algebraic groups.
If (U) is...

P-Groups as Rational Points of Unipotent Groups

This is an interesting question about the relationship between finite p-groups and connected unipotent algebraic groups over finite fields.

Background

Let's first clarify what we're looking for:

• A finite p-group is a group where every element has order a power of p
• A unipotent algebraic group is (roughly) an algebraic group that can be re...

<think> We are given a question from MathOverflow: "Is it true that every finite p-group can be realized as the group of rational points over 𝔽_p of some connected unipotent algebraic group defined over 𝔽_p? For abelian p-groups, the answer is yes via Witt vectors, but is it true in general?"

We need to answer comprehensively and accurately.

First, recall that a p-group is a group in which ev...

This is an excellent question that connects the theory of finite p-groups with the theory of linear algebraic groups. The answer is yes, and the proof is a beautiful application of the cohomology of algebraic groups.

Summary Answer

Yes, it is true that every finite p-group can be realized as the group of -rational points of some connected unipotent algebraic group defined over $...

This is a very interesting question about the relationship between finite p-groups and unipotent algebraic groups. The answer is no - not every finite p-group can be realized as the group of rational points of a connected unipotent algebraic group over .

The Key Obstruction

The main obstruction comes from the fact that if is a connected unipotent algebraic group over $\m...

No.

Abelian (p)-groups do occur as (U(\mathbf F _p)) (for instance via the truncated Witt–vector groups), but as soon as the group is non-abelian there are obstructions. The easiest one to formulate is the following.

Fact (M. Lazard, 1954). Let (U) be a connected unipotent algebraic group defined over a field (k) of characteristic (p>0), let (\dim U=d), and put
(G=U(k)).
Th...