# Status of Donovan's conjecture

Peter Donovan

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

## Donovan's conjecture by $p$-group

In the following, the column headed Donovan's conjecture indicates whether the conjecture is known over $k$ or $\mathcal{O}$.

$p$-groups Donovan's conjecture Puig's conjecture References Notes
Cyclic $p$-groups $\mathcal{O}$ Yes [Li96]
Abelian $2$-groups $P$ such that ${\rm Aut}(P)$ is a $2$-group $\mathcal{O}$ Yes All blocks are nilpotent
$C_2 \times C_2$ $\mathcal{O}$ Yes [CEKL11] Donovan's conjecture without CFSG, Puig using CFSG
Abelian $2$-groups $k$ No [EL18b]
$C_3 \times C_3$ No No [Ko03] Puig's conjecture known for principal blocks
Dihedral $2$-groups $k$ No [Er87]
Semidihedral $2$-groups $k$ No [Er88c], [Er90b]
Generalised quaternion $2$-groups No No [Er88a], [Er88b] Donovan's conjecture over $k$ known if $l(B) \neq 2$
Minimal nonabelian $2$-groups $\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle$ $\mathcal{O}$ No [EKS12] Additional assumptions on $\mathcal{O}$, which may not be necessary