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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.
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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 $mathcal{F}$.
We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C wr F$, where $C$ is a finite group and $F$ a non-abelian free group.
Let $m$ be a positive integer and let $Omega$ be a finite set. The $m$-closure of $Gle$Sym$(Omega)$ is the largest permutation group on $Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $Omega^m$. The exact formula for the $m$-closure of the wreath product in product action is given. As a corollary, a sufficient condition is obtained for this $m$-closure to be included in the wreath product of the $m$-closures of the factors.
Let $varphi: {mathbb P}^1 longrightarrow {mathbb P}^1$ be a rational map of degree greater than one defined over a number field $k$. For each prime ${mathfrak p}$ of good reduction for $varphi$, we let $varphi_{mathfrak p}$ denote the reduction of $varphi$ modulo ${mathfrak p}$. A random map heuristic suggests that for large ${mathfrak p}$, the proportion of periodic points of $varphi_{mathfrak p}$ in ${mathbb P}^1({mathfrak o}_k/{mathfrak p})$ should be small. We show that this is indeed the case for many rational functions $varphi$.
It is an open problem whether definability in Propositional Dynamic Logic (PDL) on forests is decidable. Based on an algebraic characterization by Bojanczyk, et. al.,(2012) in terms of forest algebras, Straubing (2013) described an approach to PDL based on a k-fold iterated distributive law. A proof that all languages satisfying such a k-fold iterated distributive law are in PDL would settle decidability of PDL. We solve this problem in the case k=2: All languages recognized by forest algebras satisfying a 2-fold iterated distributive law are in PDL. Furthermore, we show that this class is decidable. This provides a novel nontrivial decidable subclass of PDL, and demonstrates the viability of the proposed approach to deciding PDL in general.
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