# Difference between revisions of "Reductions"

## Donovan's conjecture

### $k$-Donovan conjecture

By [Kü95] it suffices to consider blocks of finite groups that are generated by the defect groups, i.e., the defect groups are contained in no proper normal subgroup.

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

Eisele in [Ei18] proved the analogue of [Kü95] for the $\mathcal{O}$-Donovan conjecture, so it suffices to consider blocks of finite groups that are generated by the defect groups.

By [EL20] in order to verify the $\mathcal{O}$-Donovan conjecture for a $p$-group $P$ it suffices to check it for blocks of finite groups $G$ with defect group $D \cong P$ and no proper normal subgroup $N \triangleleft G$ such that $G=C_D(D \cap N)N$.

$P$ 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].