A Note on Automorphisms of the Infinite-Dimensional Hypercube Graph

Mark Pankov

Abstract


We define the infinite-dimensional hypercube graph $H_{{\aleph}_{0}}$ as a graph whose vertex set is formed by the so-called singular subsets of ${\mathbb Z}\setminus\{0\}$. This graph is not connected, but it has isomorphic connected components. We show that the restrictions of its automorphisms to the connected components  are induced by permutations on ${\mathbb Z}\setminus\{0\}$ preserving the family of singular subsets. As an application, we describe the automorphism group of the connected components.


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