Status of Donovan's conjecture
Contents
Donovan's conjecture by [math]p[/math]-group
In October 2026 OpenAI released a claimed proof of Donovan's conjecture, with different manuscripts for [math]k[/math] ([OpenAI26a]) and [math]\mathcal{O}[/math] ([OpenAI26b]), using the classification of finite simple groups. As of 8/10/2026 we are still attempting to check/understand the proof, and this is likely to take some time (especially since we are simultaneously interested in the claimed proof of Alperin's weight conjecture!). Puig's conjecture is open, and the OpenAI manuscripts make no claims on this. Below is an account of progress made before or independently of the OpenAI proof.
In the following, the column headed Donovan's conjecture indicates whether the conjecture is known over [math]k[/math] or [math]\mathcal{O}[/math].
| [math]p[/math]-groups | Donovan's conjecture | Puig's conjecture | References | Notes |
|---|---|---|---|---|
| Cyclic [math]p[/math]-groups | [math]\mathcal{O}[/math] | Yes | [Li96] | |
| [math]C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | Yes | [CEKL11] | Donovan's conjecture without CFSG, Puig using CFSG |
| Abelian [math]2[/math]-groups | [math]\mathcal{O}[/math] | No | [EEL18] | |
| Abelian [math]3[/math]-groups | No | No | [Ko03] | Puig's conjecture known for principal blocks |
| Dihedral [math]2[/math]-groups | [math]\mathcal{O}[/math] | No | [Er87], [EEKLS26 | Full result over [math]\mathcal{O}[/math] not formally stated in literature but see discussion in [EEKLS26] |
| Semidihedral [math]2[/math]-groups | [math]k[/math] | No | [Er88c], [Er90b] | |
| [math]Q_8[/math] | [math]\mathcal{O}[/math] | No | [Er88a], [Er88b], [HKL07], [Ei16] | |
| [math]Q_8 \times C_{2^n}[/math] | [math]\mathcal{O}[/math] | No | [EL20] | |
| [math]Q_8 \times Q_8[/math] | [math]\mathcal{O}[/math] | No | [EL20] | |
| Generalised quaternion [math]2[/math]-groups | [math]\mathcal{O}[/math] | No | [Er88a], [Er88b], [Ei16], [EEKLS26] | |
| Minimal nonabelian [math]2[/math]-groups of the form [math]\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math] | [math]\mathcal{O}[/math] | No | [EKS12] | |
| Metacyclic noncyclic [math]2[/math]-groups of nonmaximal class | [math]\mathcal{O}[/math] | No | [CG12], [Sa12b] | All blocks nilpotent |
| Suzuki 2-groups | [math]\mathcal{O}[/math] | No | [Ea24] | |
| [math]p_+^{1+2}[/math] for [math]p \geq 5[/math] | [math]\mathcal{O}[/math] | No | [AE23] | |
| [math]C_{2^n} \wr C_2[/math] | Principal blocks ([math]\mathcal{O}[/math]) | Principal blocks ([math]\mathcal{O}[/math]) | [KoLaSa23] |
Donovan's conjecture by class of group or block
In the table, the column headed Donovan's conjecture indicates whether the conjecture is known over [math]k[/math] or [math]\mathcal{O}[/math].
Note that knowing the [math]\mathcal{O}[/math]-Donovan conjecture or Puig's conjecture for blocks for a class of groups does not necessarily mean that the [math]\mathcal{O}[/math]-lifts or source algebras of the [math]k[/math]-Morita equivalence classes involved are known. This is only known for elements of the Morita equivalence class which occur as blocks of groups in that class.
| Groups | Blocks | Donovan's conjecture | Puig's conjecture | References | Notes |
|---|---|---|---|---|---|
| [math]p[/math]-solvable groups | All | [math]\mathcal{O}[/math] | Yes | Over [math]k[/math] by [Ku81], Puig's conjecture by [Pu09] | See [Li18d,10.6.2] |
| Symmetric groups | All | [math]\mathcal{O}[/math] | Yes | Over [math]k[/math] by [Sc91], Puig's conjecture by [Pu94] | |
| Double covers of symmetric groups | All | [math]\mathcal{O}[/math] | Yes | [Ke96] | |
| Alternating groups and their double covers | All | [math]\mathcal{O}[/math] | Yes | [Ke02], [Ke96] | |
| [math]GL_n(q)[/math] for fixed [math]q[/math] | Unipotent blocks | [math]\mathcal{O}[/math] | Yes | Over [math]k[/math] by [Jo96], Puig's conjecture by [Ke01] | |
| Classical groups | Unipotent blocks for linear primes | [math]\mathcal{O}[/math] | Yes | [HK00], [HK05] | Detailed results beyond those stated here |
| Weyl groups of type [math]B, D[/math] | All | [math]\mathcal{O}[/math] | Yes | [Ke00] | |
| Arbitrary groups | Blocks with trivial intersection defect groups | [math]\mathcal{O}[/math] | No | [AE04] |
Weak Donovan conjecture
As described in [Dü04] the Weak Donovan conjecture is equivalent to bounding the dimensions of the Ext spaces between simple modules and bounding the Loewy length. See [GT19] and [Sh20] for progress on the former problem.
