It is shown that a countable symmetric multiplicative subgroup G = −H ∪ H with H ⊂ R * + is the group of self-similarities of a Gaussian-Kronecker flow if and only if H is additively Q-independent. In particular, a real number s = ±1 is a scale of self-similarity of a Gaussian-Kronecker flow if and only if s is transcendental. We also show that each countable symmetric subgroup of R * can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foiaş-Stratila property.
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