# Reductions

## Donovan's conjecture

### $k$-Donovan conjecture

Several reductions were achieved in [Du04], but these have been subsumed in later work.

$P$ abelian: To show the $k$-Donovan conjecture for abelian $p$-groups, it suffices to verify the Weak Donovan conjecture for blocks of quasisimple groups with abelian defect groups. We may further assume that the centre of the group is a $p'$-group. See [EL18b], [FK18].

### $\mathcal{O}$-Donovan conjecture

$P$ abelian: To show the $\mathcal{O}$-Donovan conjecture for abelian $p$-groups, it suffices to verify the Weak Donovan conjecture for blocks of quasisimple groups with abelian defect groups. We may further assume that the centre of the group is a $p'$-group. See [EEL18], [FK18].

## Weak Donovan conjecture

For arbitrary $p$-groups, it suffices to check the conjecture for blocks of quasisimple groups with centre of order not divisible by $p$. See [Du04].