Reductions

This page will contain descriptions of reduction techniques and results.

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].

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].