A note on certain permutation groups in the infinite dimensional rotation group. (English) Zbl 0611.60013

Obata, Nobuaki

Nagoya Math. J. 109, 91-107 (1988).

Let \({\mathbb{N}}\) be the set of all natural numbers and Aut(\({\mathbb{N}})\) the group of all permutations of \({\mathbb{N}}\). Let \({\mathcal G}\) be the set of all permutations \(g\in Aut({\mathbb{N}})\) such that \[ \lim _{N\to \infty}(1/N)| \{1\leq n\leq N;\quad g(n)>N\}| =0, \] where \(| \cdot |\) denotes the cardinality. Then \({\mathcal G}\) becomes a subgroup of Aut(\({\mathbb{N}})\) and is called the Lévy group.
In the paper the author discusses the following topics: (1) characterization of \({\mathcal G}\) in terms of the density of natural numbers; (2) characterization of \({\mathcal G}\) in terms of a certain functional on the Banach space \(\ell ^{\infty}\); (3) invariance of the Lévy Laplacian under \({\mathcal G}\).

MSC:

60D05	Geometric probability and stochastic geometry
20B27	Infinite automorphism groups
20P05	Probabilistic methods in group theory

Keywords:

group of all permutations; Lévy group; density of natural numbers; Lévy Laplacian

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