One parameter subgroup

A one-parameter subgroup of a Lie group $G$ is a smooth group homomorphism $\gamma :(\mathbb{R},+)\to G$: it satisfies $\gamma (s+t)=\gamma (s)\gamma (t)$, with $\gamma (0)=e$ and $\gamma (-t)=\gamma (t{)}^{-1}$. Every such map has the form $\gamma (t)=\exp (tX)$ for a unique $X={\gamma }^{\prime }(0)$ in the Lie algebra of $G$. Thus its initial tangent vector determines the whole motion, for all real times. The map need not be injective; the word subgroup describes its image. See MIT 18.745, Lecture 7, Proposition 7.1.