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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 $mathcal{F}$.
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
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 s
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 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 tru