On orientation-preserving automorphisms of Hamiltonian cycles in the <Emphasis Type="Italic">N</Emphasis>-dimensional Boolean cube

We obtain an upper bound for the order of the group of orientation-preserving automorphisms of a Hamiltonian cycle in the Boolean n-cube. We prove that the existence of a Hamiltonian cycle with the order of the group of orientation-preserving automorphisms attaining this upper bound is equivalent to the existence of a Hamiltonian cycle with an additional condition on the graph of orbits of a fixed automorphism of the n-cube.