Products in Fusion Systems

We revisit the notion of a product of a normal subsystem with a $p$-subgroup as defined by Aschbacher. In particular, we give a previously unknown, more transparent construction.Comment: 18 pages; minor revisions; to appear in the Journal of Algebr

Subcentric linking systems

20 pagesPeer reviewedPreprin

Products of partial normal subgroups

We show that the product of two partial normal subgroups of a locality (in the sense of Chermak) is again a partial normal subgroup. This generalizes a theorem of Chermak and fits into the context of building a local theory of localities.Comment: 10 pages; compared to the first version some more background on localities included; this version is accepted to the Robert Steinberg Memorial Issue of the Pacific Journal of Mathematic

A characterization of saturated fusion systems over abelian 2-groups

Given a saturated fusion system $\mathcal{F}$ over a $2$-group $S$, we prove that $S$ is abelian provided any element of $S$ is $\mathcal{F}$-conjugate to an element of $Z(S)$. This generalizes a Theorem of Camina--Herzog, leading to a significant simplification of its proof. More importantly, it follows that any $2$-block $B$ of a finite group has abelian defect groups if all $B$-subsections are major. Furthermore, every $2$-block with a symmetric stable center has abelian defect groups.Comment: 4 pages. One corollary added in the updated version. Accepted to appear in Adv. Mat

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.Comment: 42 pages, accepted to Trans. Amer. Math. So

Centralizers of normal subgroups and the Z∗Z^*-Theorem

Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$-subgroup $S$, the centre of $G/O_{p^\prime}(G)$ is determined by the fusion system $\mathcal{F}_S(G)$. Building on these results we show a statement that seems a priori more general: For any normal subgroup $H$ of $G$ with $O_{p^\prime}(H)=1$, the centralizer $C_S(H)$ is expressed in terms of the fusion system $\mathcal{F}_S(H)$ and its normal subsystem induced by $H$.Comment: 3 pages; to appear in the Journal of Algebr