We characterize monothetic topological groups among one-parametric topological groups. In particular, we prove that a topological group of weight $<cov(M)$ is monothetic if and only if it is not isomorphic to the real line. This implies that the real line is a unique non-monothetic metrizable one-parametric topological group. Also we prove that the real line with the Bohr topology as a unique non-monothetic totally bounded $N$-scaled topological group.

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