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Ellen Henke

Publications and source records attributed to Ellen Henke.

At least 19 recordsLinked to original sources
arXiv2026

Large subgroups of fusion systems and localities

Saturated fusion systems are categories modeling properties of conjugacy of p-subgroups in finite groups. It was shown by Chermak that they correspond nicely to group-like structures called localities. In this paper we start to explore how concepts and results from a program of Meierfrankenfeld, Stellmacher and Stroth, aiming to reprove and generalize parts of the classification of the finite simple groups, translate to fusion systems and localities. Central in the program is the notion of a large $p$-subgroup. The presence of a large $p$-subgroup in a finite group turns out to be strong enough information to nearly classify the entire $p$-local structure, while also accommodating a very large class of groups of interest including many groups of Lie type in defining characteristic $p$. Utilizing the group-theoretical definition of a large $p$-subgroup as a blueprint, we define large subgroups of fusion systems and localities. We then analyze how the three definitions relate to each other, showing in particular that the newly defined notions behave well under the correspondence between saturated fusion systems and localities with certain properties. We further proceed with an example of how classification results from the program of Meierfrankenfeld et.al. translate to fusion systems and localities. In more detail, we give a new characterization of the $2$-fusion system of $\operatorname{Aut}(\operatorname{G}_2(3))$ following the strategy in a paper of Meierfrankenfeld and Stroth, where the group $\operatorname{Aut}(\operatorname{G}_2(3))$ is characterized in a similar fashion.

math.GR↗
arXiv2025

Components and realizability of fusion systems

For $p\in\{2,3\}$ it is known that a saturated $p$-fusion system is realizable if and only if each of its components is realizable by a finite simple group. For primes $p\geq 5$ this is false. Building on work of Broto, M{\o}ller, Oliver and Ruiz, we show however that a fusion system $\mathcal{F}$ is realizable if and only if for each of its components $\mathcal{C}$ there exists a realizable subnormal subsystem $\mathcal{E}$ of $\mathcal{F}$ with $O^{p^\prime}(\mathcal{E})=\mathcal{C}$.

math.GR↗
arXiv2022

Fusion systems and localities -- a dictionary

Linking systems were introduced to provide algebraic models for $p$-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system $\mathcal{F}$ corresponds to a group-like structure called a locality. Given such a locality $\mathcal{L}$, we prove that there is a one-to-one correspondence between the partial normal subgroups of $\mathcal{L}$ and the normal subsystems of the fusion system $\mathcal{F}$. This is then used to obtain a kind of dictionary, which makes it possible to translate between various concepts in localities and corresponding concepts in fusion systems. As a byproduct, we obtain new proofs of many known theorems about fusion systems and also some new results. For example, we show in this paper that, in any saturated fusion system, there is a sensible notion of a product of normal subsystems.

math.GR↗
arXiv2022

Kernels of Localities

We state a sufficient condition for a fusion system to be saturated. This is then used to investigate localities with kernels, i.e. localities which are (in a particular way) extensions of groups by localities. As an application of these results, we define and study certain products in fusion systems and localities, thus giving a new method to construct saturated subsystems of fusion systems.

math.GR↗
arXiv2022

Some products in fusion systems and localities

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 $\mathcal{F}$ is a saturated fusion system over a $p$-group $S$. If $\mathcal{E}$ is a normal subsystem of $\mathcal{F}$ over $T\leq S$, and $\mathcal{D}$ is a subnormal subsystem of $N_{\mathcal{F}}(T)$ over $R\leq S$, then there is a subnormal subsystem $\mathcal{E}\mathcal{D}$ of $\mathcal{F}$ over $TR$, which plays the role of a product of $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{F}$. If $\mathcal{D}$ is normal in $N_{\mathcal{F}}(T)$, then $\mathcal{E}\mathcal{D}$ is normal in $\mathcal{F}$. It is shown along the way that the subsystem $\mathcal{E}\mathcal{D}$ is closely related to a naturally arising product in certain localities attached to $\mathcal{F}$.

math.GR↗
arXiv2022

On Wielandt's Join Theorem for fusion systems and localities

Saturated fusion systems are categories generalizing important aspects of conjugacy of $p$-subgroups in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group $G$ generate a subnormal subgroup of $G$. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.

math.GR↗
arXiv2022

Punctured groups for exotic fusion systems

The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson-Solomon exotic fusion system at the prime $2$ has a punctured group, while the others do not. As for exotic fusion systems at odd primes $p$, we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order $p$ is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial $p$-groups of order $p^3$.

math.GR↗
arXiv2021

Normalizers and centralizers of subnormal subsystems of fusion systems

Every saturated fusion system corresponds to a group-like structure called a regular locality. In this paper we study (suitably defined) normalizers and centralizers of partial subnormal subgroups of regular localities. This leads to a reasonable notion of normalizers and centralizers of subnormal subsystems of fusion systems.

math.GR↗
arXiv2021

Commuting partial normal subgroups and regular localities

In this paper, important concepts from finite group theory are translated to localities, in particular to linking localities. Here localities are group-like structures associated to fusion systems which were introduced by Chermak. Linking localities (by Chermak also called proper localities) are special kinds of localities which correspond to linking systems. Thus they contain the algebraic information that is needed to study $p$-completed classifying spaces of fusion systems as generalizations of $p$-completed classifying spaces of finite groups. Because of the group-like nature of localities, there is a natural notion of partial normal subgroups. Given a locality $\mathcal{L}$ and a partial normal subgroup $\mathcal{N}$ of $\mathcal{L}$, we show that there is a largest partial normal subgroup $\mathcal{N}^\perp$ of $\mathcal{L}$ which, in a certain sense, commutes elementwise with $\mathcal{N}$ and thus morally plays the role of a "centralizer" of $\mathcal{N}$ in $\mathcal{L}$. This leads to a nice notion of the generalized Fitting subgroup $F^*(\mathcal{L})$ of a linking locality $\mathcal{L}$. Building on these results we define and study special kinds of linking localities called regular localities. It turns out that there is a theory of components of regular localities akin to the theory of components of finite groups. The main concepts we introduce and work with in the present paper (in particular $\mathcal{N}^\perp$ in the special case of linking localities, $F^*(\mathcal{L})$, regular localities and components of regular localities) were already introduced and studied in a preprint by Chermak. However, we give a different and self-contained approach to the subject where we reprove Chermak's theorems and also show several new results.

math.GR↗
arXiv2021

Normalizers and centralizers of $p$-subgroups in normal subsystems of fusion systems

Suppose $\mathcal{E}$ is a normal subsystem of a saturated fusion system $\mathcal{F}$ over $S$. If $X\leq S$ is fully $\mathcal{F}$-normalized, then Aschbacher defined a normal subsystem $N_{\mathcal{E}}(X)$ of $N_{\mathcal{F}}(X)$. In this short note we revisit and generalize this result using the theory of localities. Our more general approach leads in particular to a normal subsystem $C_{\mathcal{E}}(X)$ of $C_{\mathcal{F}}(X)$ for every $X\leq S$ which is fully $\mathcal{F}$-centralized.

math.GR↗
arXiv2021

Fusion systems with Benson-Solomon components

The Benson-Solomon systems comprise a one-parameter family of simple exotic fusion systems at the prime $2$. The results we prove give significant additional evidence that these are the only simple exotic $2$-fusion systems, as conjectured by Solomon. We consider a saturated fusion system $\mathcal{F}$ having an involution centralizer with a component $\mathcal{C}$ isomorphic to a Benson-Solomon fusion system, and we show under rather general hypotheses that $\mathcal{F}$ cannot be simple. Furthermore, we prove that if $\mathcal{F}$ is almost simple with these properties, then $\mathcal{F}$ is isomorphic to the next larger Benson-Solomon system extended by a group of field automorphisms. Our results are situated within Aschbacher's program to provide a new proof of a major part of the classification of finite simple groups via fusion systems. One of the most important steps in this program is a proof of Walter's Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walter's Theorem to treat the general Benson-Solomon component problem under the assumption that each component of an involution centralizer in $\mathcal{F}$ is on the list of currently known quasisimple $2$-fusion systems.

math.GR↗
arXiv2020

Extensions of homomorphisms between localities

We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal{F}$ depends only on $\mathcal{F}$ as long as the object set of the linking system is $\mathrm{Aut}(\mathcal{F})$-invariant. This was known to be true for linking systems in Oliver's definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this paper. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

math.GR↗
arXiv2018

Extensions of the Benson-Solomon fusion systems

The Benson-Solomon systems comprise the only known family of simple saturated fusion systems at the prime two that do not arise as the fusion system of any finite group. We determine the automorphism groups and the possible almost simple extensions of these systems and of their centric linking systems.

math.GR↗
arXiv2018

Centralizers of normal subsystems revisited

In this paper we revisit two concepts which were originally introduced by Aschbacher and are crucial in the theory of saturated fusion systems: Firstly, we give a new approach to defining the centralizer of a normal subsystem. Secondly, we revisit the construction of the product of two normal subsystems which centralize each other.

math.GR↗
arXiv2018

Subcentric linking systems

Linking systems are crucial for studying the homotopy theory of fusion systems, but are also of interest from an algebraic point of view. We propose a definition of a linking system associated to a saturated fusion system which is more general than the one currently in the literature and thus allows a more flexible choice of objects of linking systems. More precisely, we define subcentric subgroups of fusion systems in a way that every quasicentric subgroup of a saturated fusion system is subcentric. Whereas the objects of linking systems in the current definition are always quasicentric, the objects of our linking systems only need to be subcentric. We prove that, associated to each saturated fusion system $\mathcal{F}$, there is a unique linking system whose objects are the subcentric subgroups of $\mathcal{F}$. Furthermore, the nerve of such a subcentric linking system is homotopy equivalent to the nerve of the centric linking system associated to $\mathcal{F}$. We believe that the existence of subcentric linking systems opens a new way for a classification of fusion systems of characteristic $p$-type. The various results we prove about subcentric subgroups give furthermore some evidence that the concept is of interest for studying extensions of linking systems and fusion systems.

math.GR↗
arXiv2016

Control of Fusion by Abelian Subgroups of the Hyperfocal Subgroup

We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most 4 if p = 2. For odd p, this has implications for mod p group cohomology.

math.GR↗