A cohomological characterization of nilpotent fusion systems

TL;DR

This paper introduces a cohomological criterion to determine when a fusion system is nilpotent, linking algebraic properties with topological invariants.

Contribution

It offers a novel nilpotency criterion for fusion systems based on the vanishing of their cohomology with twisted coefficients.

Findings

Cohomology with twisted coefficients characterizes nilpotent fusion systems.

Provides a practical criterion for identifying nilpotency.

Connects algebraic and topological aspects of fusion systems.

Abstract

We provide a nilpotency criterion for fusion systems in terms of the vanishing of its cohomology with twisted coefficients.

Equations

$$1\to N\to G\to S$$

$$G$$

$$\forall P\le S\text{, }\forall \phi \in {\mathrm{Hom}}_{\mathcal{F}}(P,S)\text{, }\forall p\in P\text{, }\forall m\in M:\phi (p)\cdot m=p\cdot m.$$

$$\mathfrak{foc}(\mathcal{F})=⟨[P,{\mathrm{Aut}}_{\mathcal{F}}(P)]\text{, }P\le S⟩.$$

$$\mathfrak{hyp}(\mathcal{F})=⟨[P,O^p({\mathrm{Aut}}_{\mathcal{F}}(P))]\text{, }P\le S⟩.$$

$$H^n({\mathcal{F}}^c;M)=\{z\in H^n(S;M)|\forall P\in {\mathcal{F}}^c\text{, }\forall \phi \in {\mathrm{Hom}}_{\mathcal{F}}(P,S):res(z)={\phi }^{\ast }(z)\}.$$

$$H^n(\mathcal{F};M)=\{z\in H^n(S;M)|\forall P\le S\text{, }\forall \phi \in {\mathrm{Hom}}_{\mathcal{F}}(P,S):res(z)={\phi }^{\ast }(z)\}.$$

$$H^{\ast }({\mathcal{F}}^c;M)\cong H^{\ast }(B\mathcal{F};M),$$

$$H^n({\mathcal{F}}_S(S{)}^c;M)=H^n(S;M).$$

$$\pi \to B{O}^p(\mathcal{F})\to B\mathcal{F},$$

$$H^1({\mathcal{F}}^c;M)\cong H^1(B\mathcal{F};M)\cong H^1(B{O}^p(\mathcal{F});{\mathbb{F}}_p)=0,$$

$$0=H^n({\mathcal{F}}^c;M)\cong H^n(B{O}^p(\mathcal{F});{\mathbb{F}}_p)$$

$$B{O}^p(G)\to BG\to BG/O^p(G),$$

Full text

Authors partially supported by MEC grants MTM2013-41768-P and MTM2016-78647-P and Junta de Andalucía grant FQM-213. Second author supported by Polish National Science Centre grant 2016/21/P/ST1/03460 within the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778.

1. Introduction

Let $G$ be a finite group and let $p$ be a prime. Then $G$ is said to be $p$-nilpotent if a Sylow $p$-subgroup has a complement, i.e., if there exists a split short exact sequence

[TABLE]

with $S\in\operatorname{Syl}\nolimits_{p}(G)$. It turns out that this property can be characterized solely in terms of the fusion system of $G$ over $S$. This terminology was introduced in [6] and it is straightforward that

[TABLE]

where $\mathcal{F}_{S}(G)$ denotes the the fusion system of $G$ over $S$. Consequently, a fusion system $\mathcal{F}$ over the $p$-group $S$ is termed nilpotent if $\mathcal{F}=\mathcal{F}_{S}(S)$. Already several authors have provided fusion system counterparts to characterizations of $p$-nilpotency for finite groups, see [2], [3], [7], [9], [10], [11], [13] and [14]. In this work, we prove the fusion system version of a $p$-nilpotency criterion from the late $60$’s due to Wong [16] and Hoechsmann, Roquette and Zassenhaus [12].

Theorem 1.1**.**

Let $\mathcal{F}$ be a fusion system. Then the following are equivalent:

1. (1)

$\mathcal{F}$* is nilpotent.* 2. (2)

For each $\mathbb{F}_{p}[\mathcal{F}]$-module $M$, if $H^{m}(\mathcal{F}^{c};M)=0$ for some $m>0$ then $H^{n}(\mathcal{F}^{c};M)=0$ for every $n>0$.

Here, a $\mathbb{F}_{p}[\mathcal{F}]$-module is a finitely generated $\mathbb{F}_{p}[S]$-module that is $\mathcal{F}$-invariant, and $H^{*}(\mathcal{F}^{c};M)$ is twisted cohomology over $\mathcal{F}^{c}$, i.e., over the $\mathcal{F}$-centric subgroups. See Definitions 2.1 and 2.2 for full details. We give a topological proof of Theorem 1.1 via the classifying space $B\mathcal{F}$ of $\mathcal{F}$ [8]. Recall that, in the terminology of [6], $B\mathcal{F}\simeq|\mathcal{L}|^{\wedge}_{p}$, where $\mathcal{L}$ is the unique centric linking system associated to $\mathcal{F}$ and $(\cdot)^{\wedge}_{p}$ denotes $p$-completion in the sense of Bousfield and Kan [4].

The original version of Theorem 1.1 was equivalently stated in terms of Tate’s cohomology. Moreover, its proof resorted to dimension shifting. This approach is not suitable here as there are not enough acyclic modules. For instance, if $B\mathcal{F}$ is simply connected, then, for each $\mathbb{F}_{p}[\mathcal{F}]$-module $M$, $H^{*}(\mathcal{F}^{c};M)=H^{*}(\mathcal{F};M)$ is cohomology with trivial coefficients. Hence, it will not vanish but in trivial cases.

Notation: Throughout this work, by fusion system we mean a saturated fusion system. The unacquainted reader may find an explanation of this terminology and general background on fusion systems in [1].

2. Proof of the theorem

We start introducing modules for fusion systems and their cohomology:

Definition 2.1**.**

Let $\mathcal{F}$ be a fusion system over the finite $p$-group $S$. An $\mathbb{F}_{p}[\mathcal{F}]$-module is a finitely generated $\mathbb{F}_{p}[S]$-module $M$ which is $\mathcal{F}$-invariant, i.e., such that:

[TABLE]

It is clear that an $\mathbb{F}_{p}[\mathcal{F}]$-module is the same thing as a finitely generated $\mathbb{F}_{p}[S/\mathfrak{foc}(\mathcal{F})]$-module, where $\mathfrak{foc}(\mathcal{F})$ is the focal subgroup of $\mathcal{F}$:

[TABLE]

By inflation, every $\mathbb{F}_{p}[\mathcal{F}]$-module is also a $\mathbb{F}_{p}[S/\mathfrak{hyp}(\mathcal{F})]$-module, where $\mathfrak{hyp}(\mathcal{F})$ is the hyperfocal subgroup of $\mathcal{F}$:

[TABLE]

Definition 2.2** ([15, Definition 2.3]).**

Let $\mathcal{F}$ be a saturated fusion system over the finite $p$-group $S$ and let $M$ be an $\mathbb{F}_{p}[\mathcal{F}]$-module. For each $n\geq 0$, define the twisted cohmology group $H^{n}(\mathcal{F}^{c};M)$ as the $\mathcal{F}^{c}$-stable elements:

[TABLE]

Here, $\varphi^{*}:H^{n}(S;M)\to H^{n}(P;M)$ is the homomorphism induced in cohomology by $\varphi$, and $res=\iota^{*}$ for the inclusion $\iota\colon P\hookrightarrow S$. By Alperin’s Fusion Theorem, if the action of $S/\mathfrak{hyp}(\mathcal{F})$ on $M$ is trivial, then $H^{n}(\mathcal{F}^{c};M)$ coincide with the $\mathcal{F}$-stable elements

[TABLE]

In general, the abelian group $H^{n}(\mathcal{F}^{c};M)$ may be recovered via topology as the cohomology of the classifying space of $\mathcal{F}$ [15, Corollary 5.4]:

[TABLE]

where $\pi_{1}(B\mathcal{F})=S/\mathfrak{hyp}(\mathcal{F})$ by [5, Theorem B]. Now we are ready to prove the main theorem.

Proof of Theorem 1.1.

For the implication $(1)\Rightarrow(2)$, note that:

[TABLE]

From here it is enough to follow the group theoretical proof, see [16, Theorem 1] or [12, Proposition 1a]. For the reverse implication, set $\pi=S/\mathfrak{hyp}(\mathcal{F})$ and consider the universal covering space [5, Theorem 4.4],

[TABLE]

where $O^{p}(\mathcal{F})\leq\mathcal{F}$ is the unique $p$-power index fusion subsystem of $\mathcal{F}$ over $\mathfrak{hyp}(\mathcal{F})\leq S$. For $M=\mathbb{F}_{p}[\pi]$ with $\pi$ acting by left multiplication, we get

[TABLE]

as $BO^{p}(\mathcal{F})$ is simply connected. Hence, by hypothesis:

[TABLE]

for all $n\geq 1$. As $BO^{p}(\mathcal{F})$ is a $p$-complete space [6, Proposition 1.11], it must be contractible [4, I.5.5]. Hence, $B\mathcal{F}\simeq K(\pi,1)$, $\mathfrak{hyp}(\mathcal{F})=1$ and $\mathcal{F}=\mathcal{F}_{S}(S)$. ∎

Remark 2.3**.**

This argument provides an alternative proof of the implication $(2)\Rightarrow(1)$ for finite groups. Namely, consider the following fibration of classifying spaces of finite groups,

[TABLE]

and then $p$-complete it [4, II.5.1].