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