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]]]. | + | By [[References#K|[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. |
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| + | Several reductions were achieved in [[References#D|[Du04]]], but these have been subsumed in later work. | ||
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| + | '''<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]]]. | ||
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| + | === <math>\mathcal{O}</math>-Donovan conjecture === | ||
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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. | ||
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| + | 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>. | ||
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| + | '''<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]]]. | ||
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| + | '''<math>3_+^{1+2}</math>:''' To show the <math>\mathcal{O}</math>-Donovan conjecture for <math>3_+^{1+2}</math>, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with defect groups <math>3_+^{1+2}</math>. See [[References#A|[AE23]]]. | ||
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| + | == Weak Donovan conjecture == | ||
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| + | 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]]]. | ||
Latest revision as of 08:34, 3 October 2023
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].
[math]3_+^{1+2}[/math]: To show the [math]\mathcal{O}[/math]-Donovan conjecture for [math]3_+^{1+2}[/math], it suffices to verify the Weak Donovan conjecture for blocks of quasisimple groups with defect groups [math]3_+^{1+2}[/math]. See [AE23].
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].
