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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.
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 th
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
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
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 normali
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 an