# Difference between revisions of "Reductions"

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=== <math>k</math>-Donovan conjecture === | === <math>k</math>-Donovan conjecture === | ||

− | '''<math>P</math> abelian:''' To show the <math>k</math>-Donovan conjecture for abelian <math>p</math>-groups, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with abelian defect groups. See [[References|[EL18b], [FK18]]]. | + | Several reductions were achieved in [[References|[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 [[Statements of conjectures#WeakDonovan conjecture|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 [[References|[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 [[Statements of conjectures#WeakDonovan conjecture|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 [[References|[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 [[References|[Du04]]]. |

## Revision as of 17:37, 23 September 2018

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].