# Difference between revisions of "Reductions"

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Eisele in [[References#E|[Ei18]]] proved the analogue of [[References#K|[Kü95]]] for the <math>\mathcal{O}</math>-Donovan conjecture, so it suffices to consider blocks of finite groups that are generated by the defect groups. | Eisele in [[References#E|[Ei18]]] proved the analogue of [[References#K|[Kü95]]] for the <math>\mathcal{O}</math>-Donovan conjecture, so it suffices to consider blocks of finite groups that are generated by the defect groups. | ||

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

'''<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#E|[EEL18]]], [[References#F|[FK18]]]. | '''<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#E|[EEL18]]], [[References#F|[FK18]]]. |

## Latest revision as of 11:43, 29 June 2020

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

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.

**[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

Eisele in [Ei18] proved the analogue of [Kü95] for the [math]\mathcal{O}[/math]-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 [math]\mathcal{O}[/math]-Donovan conjecture for a [math]p[/math]-group [math]P[/math] it suffices to check it for blocks of finite groups [math]G[/math] with defect group [math]D \cong P[/math] and no proper normal subgroup [math]N \triangleleft G[/math] such that [math]G=C_D(D \cap N)N[/math].

**[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].