# Reductions

This page will contain descriptions of reduction techniques and results.

## Contents

## Donovan's conjecture

For the statement of the conjecture click here.

### [math]k[/math]-Donovan conjecture

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

**[math]P[/math] abelian:** To show the [math]k[/math]-Donovan conjecture for abelian [math]p[/math]-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 [math]p'[/math]-group. See [EL18b], [FK18].

### [math]\mathcal{O}[/math]-Donovan conjecture

**[math]P[/math] abelian:** To show the [math]\mathcal{O}[/math]-Donovan conjecture for abelian [math]p[/math]-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 [math]p'[/math]-group. See [EEL18], [FK18].

## Weak Donovan conjecture

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