Fusion systems and localities – a dictionary

Chermak, Andrew; Henke, Ellen

doi:10.1016/j.aim.2022.108690

Cited by 4 publications

(29 citation statements)

References 19 publications

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“…Setting Δ := {𝑃 ≤ 𝑆 : 𝑃𝑂 𝑝 (L) ∈ Δ }, the triple (L, Δ, 𝑆) is also a locality (but with a possibly larger object set). It turns out that (L, Δ, 𝑆) is a linking locality if and only if (L, Δ, 𝑆) is a linking locality (compare [10,Lemma 3.28]). This flexibility in the choice of object sets makes it possible to formulate a result similar to Theorem C for regular localities.…”

Section: Definitionmentioning

confidence: 99%

“…For a saturated fusion system F with E F, it is shown, for example, in [10,Lemma 7.13(a)] that 𝐸 (E) F. Hence, it makes sense in the situation above to form the product subsystem 𝐸 (E)𝑆 0 . Moreover, part (b) of Corollary F implies the following statement: If D is a saturated subsystem of F with E D and…”

Section: Corollary Fmentioning

confidence: 99%

“…We will now look at some properties of the set 𝛿(F) introduced in Subsection 3.5. As shown in [10,Lemma 7.21], 𝛿(F) can be characterised as the set of subgroups of S containing an element of 𝐹 * (F) 𝑠 . The reader is referred to [3] or [10,Definition 7.2] for the definition of the generalised Fitting subsystem 𝐹 * (F) and the layer 𝐸 (F) of F. Recall Notation 3.21.…”

Section: Lemma 46 Let E Be a Weakly Normal Subsystem Ofmentioning

confidence: 99%

“…Assume now that (i) holds. Notice that As 𝑇 0 ∈ 𝐸 (E) 𝑠 ⊆ 𝛿(E) = Γ ⊆ Δ by [10,Lemma 7.22], it follows that 𝐺 := 𝑁 L (𝑇 0 ) is a group with 𝑁 := 𝑁 N (𝑇 0 ) 𝐺. Indeed, since (N , Γ, 𝑇) is a regular locality, N is of characteristic p. Hence, by Lemma 4.3, G is of characteristic p if and only if 𝑁 𝐺 (𝑇) is of characteristic p. By [18,Lemma 11.12], every automorphism of N leaves 𝐸 (N ) invariant, so 𝑁 L (𝑇) acts by Lemma 5.2(b) on 𝐸 (N ) via conjugation.…”

Section: Kernels Of Localitiesmentioning

confidence: 99%

“…Set F := F 𝑆 (L). By[10, Theorem E(d)], 𝐸 (E) = F 𝐸 (N )∩𝑆 (𝐸 (N )), and in particular 𝑇 0 := 𝐸 (N ) ∩ 𝑆 = 𝐸 (E) ∩ 𝑆. As (N , Γ, 𝑇) is regular (and in particular a linking locality), it follows from Theorem C that (i) and (iii) are equivalent.…”

mentioning

confidence: 94%

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Kernels of localities

Grazian

Henke²

2022

Forum of Mathematics, Sigma

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Section: Definitionmentioning

confidence: 99%

Section: Corollary Fmentioning

confidence: 99%

Section: Lemma 46 Let E Be a Weakly Normal Subsystem Ofmentioning

confidence: 99%

Section: Kernels Of Localitiesmentioning

confidence: 99%

mentioning

confidence: 94%

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Kernels of localities

Grazian

Henke²

2022

Forum of Mathematics, Sigma

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Finite localities I

Chermak

2022

Forum of Mathematics, Sigma

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This is the first of a series of papers concerning what might be thought of as ‘locally grouped spaces’, in a loose analogy with the locally ringed spaces of algebraic geometry. The spaces that we have in mind are simplicial sets that generalise the simplicial sets that underlie and determine the classifying spaces of finite (or compact) groups. If the analogy is pursued, then the role of ‘structure sheaf’ is provided by the ‘fusion systems’ associated with these spaces. Our approach here will be purely algebraic and combinatorial, so we will not be concerned with topological realisations. All of the groups to be considered will be finite; but a parallel series of papers representing some joint work with Alex Gonzalez will considerably broaden the scope.

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Some products in fusion systems and localities

Henke

2022

Proc. Amer. Math. Soc.

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The theory of saturated fusion systems resembles in many parts the theory of finite groups. However, some concepts from finite group theory are difficult to translate to fusion systems. For example, products of normal subsystems with other subsystems are only defined in special cases. In this paper the theory of localities is used to prove the following result: Suppose F \mathcal {F} is a saturated fusion system over a p p -group S S . If E \mathcal {E} is a normal subsystem of F \mathcal {F} over T ≤ S T\leq S , and D \mathcal {D} is a subnormal subsystem of N F ( T ) N_\mathcal {F}(T) over R ≤ S R\leq S , then there is a subnormal subsystem E D \mathcal {E}\mathcal {D} of F \mathcal {F} over T R TR , which plays the role of a product of E \mathcal {E} and D \mathcal {D} in F \mathcal {F} . If D \mathcal {D} is normal in N F ( T ) N_\mathcal {F}(T) , then E D \mathcal {E}\mathcal {D} is normal in F \mathcal {F} . It is shown along the way that the subsystem E D \mathcal {E}\mathcal {D} is closely related to a naturally arising product in certain localities attached to F \mathcal {F} .

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Fusion systems and localities – a dictionary

Cited by 4 publications

References 19 publications

Kernels of localities

Kernels of localities

Finite localities I

Some products in fusion systems and localities

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