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Register a new userSome products in fusion systems and localities
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Authors
Ellen Henke
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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 $mathcal{F}$ is a saturated fusion system over a $p$-group $S$. If $mathcal{E}$ is a normal subsystem of $mathcal{F}$ over $Tleq S$, and $mathcal{D}$ is a normal subsystem of $N_{mathcal{F}}(T)$ over $Rleq S$, then there is a normal subsystem $mathcal{E}mathcal{D}$ of $mathcal{F}$ over $TR$, which plays the role of a product of $mathcal{E}$ and $mathcal{D}$ in $mathcal{F}$. 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 $mathcal{F}$.
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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.
We develop a theory of semidirect products of partial groups and localities. Our concepts generalize the notions of direct products of partial groups and localities, and of semidirect products of groups.
In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.
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 Chermaks theorems and also show several new results.
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 Olivers 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.
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