A note on automorphisms of the infinite-dimensional hypercube graph (Q1953329)

Summary: 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.