? 2022It is shown that every non-trivial subgroup with one generator p of the multiplicative group of positive numbers R+ is the stabilizer of some cloud in the Gromov–Hausdorff class. Namely, this is the stabilizer both of the cloud of the geometric progression Xp?R+ and of the cloud of the discrete hedgehog Xp? over this progression. It is proved that the center ZXp? consists of a single complete space. It is proved that the center Z0,∞) (of the half-line cloud) also consists of a single complete space, but the half-line cloud stabilizer is the maximum possible – the whole group of similarities R+. It is shown that the stabilizer of any cloud is the union of isometric stabilizers of spaces from the center of this cloud.
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