= One-parameter subgroup
{title2=$\gamma_X(t)=\exp(tX)$}
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= One-parameter group
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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 $X=\gamma'(0)$ 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 https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec07.pdf[MIT 18.745, Lecture 7, Proposition 7.1].
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