Difference between revisions of "Reductions"
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=== <math>k</math>-Donovan conjecture === | === <math>k</math>-Donovan conjecture === | ||
| − | Several reductions were achieved in [[References|[Du04]]], but these have been subsumed in later work. | + | Several reductions were achieved in [[References#D|[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>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#E|[EL18b]]], [[References#F|[FK18]]]. |
=== <math>\mathcal{O}</math>-Donovan conjecture === | === <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]]]. | + | '''<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]]]. |
== Weak Donovan conjecture == | == 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]]]. | + | 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#D|[Du04]]]. |
Revision as of 22:46, 18 December 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].
