Control of transfer and weak closure in fusion systems

“…By Lemma 2.8, any saturated fusion system with essential rank 0 is constrained. The importance of being constrained is that it reduces some questions about fusion systems to questions about groups (see [9], [8] or [14] for some recent examples).…”

Section: Preliminariesmentioning

confidence: 99%

“…As with the analogue for group theory, the focal subgroup of a fusion system controls the existence of subsystems of p-power index with abelian quotient. For more detail and properties we refer the reader to [2] and [9]. In the former reference, the reader can find a definition of O p (F ).…”

Section: Extremely Sparse Fusion Sytemsmentioning

confidence: 99%

Sparse fusion systems

Glesser¹

2012

Proceedings of the Edinburgh Mathematical Society

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“…This implies D ′ ∩ Z = 1, since λ extends to D. Let F be the fusion system of B. Then Z ≤ Z(F) and it follows from [4,Lemma 4.3] that Z ∩ foc(F) = 1.…”

mentioning

confidence: 94%

On the projective height zero conjecture

Sambale

2018

Proc. Amer. Math. Soc.

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Recently, Malle and Navarro put forward a projective version of Brauer's celebrated height zero conjecture on blocks of finite groups. In this short note we show that Brauer's original conjecture implies the projective version.The following is a long-standing conjecture in representation theory of finite groups: Conjecture 1 (Brauer's height zero conjecture [2, Problem 23]). Let B be a block of a finite group with defect D. Then every irreducible character in B has height 0 if and only if D is abelian. Recently, Malle-Navarro [6] proposed the following generalization of Conjecture 1 (the case Z = 1 yields the original conjecture). An equivalent statement in terms of θ-blocks was given by Rizo [9]. Conjecture 2 (Malle-Navarro's projective height zero conjecture). Let B be a p-block of a finite group G with defect group D. Let Z be a central p-subgroup of G and let λ ∈ Irr(Z). Then every irreducible character in B lying over λ has height 0 if and only if D/Z is abelian and λ extends to D.

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Control of transfer and weak closure in fusion systems

Cited by 8 publications

References 7 publications

Fusion systems

Fusion systems

Sparse fusion systems

On the projective height zero conjecture

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