Status of Donovan's conjecture

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Peter Donovan

In this page we list cases where Donovan's conjecture is known to hold.

Donovan's conjecture by [math]p[/math]-group

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]
Abelian [math]2[/math]-groups [math]P[/math] such that [math]{\rm Aut}(P)[/math] is a [math]2[/math]-group [math]\mathcal{O}[/math] Yes All blocks are nilpotent
[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]k[/math] No [EL18b]
[math]C_3 \times C_3[/math] No No [Ko03] Puig's conjecture known for principal blocks
Dihedral [math]2[/math]-groups [math]k[/math] No [Er87]
Semidihedral [math]2[/math]-groups [math]k[/math] No [Er88c], [Er90b]
Generalised quaternion [math]2[/math]-groups No No [Er88a], [Er88b] Donovan's conjecture over [math]k[/math] known if [math]l(B) \neq 2[/math]
Minimal nonabelian [math]2[/math]-groups [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] Additional assumptions on [math]\mathcal{O}[/math], which may not be necessary