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Register a new userNormalizers and centralizers of $p$-subgroups in normal subsystems of fusion systems
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Authors
Ellen Henke
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Suppose $mathcal{E}$ is a normal subsystem of a saturated fusion system $mathcal{F}$ over $S$. If $Xleq S$ 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 $Xleq S$ which is fully $mathcal{F}$-centralized.
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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.
It has been known that the centralizer $Z_W(W_I)$ of a parabolic subgroup $W_I$ of a Coxeter group $W$ is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $mathcal{Y}$. In this paper, we study the structure of $Z_W(W_I)$ further and show that, if $I$ has no irreducible components of type $A_n$ with $2 leq n < infty$, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of $mathcal{Y}$. This property has an application to the isomorphism problem in Coxeter groups.
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 announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.
In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic $p$-group. Lastly, we note that class-preserving automorphisms of 2-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge.
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