One-parameter subgroup of a Lie group

The standard coordinate on $\mathbb{R}$ makes addition $(s,t)\mapsto s+t$ and inversion $t\mapsto -t$ smooth, so $(\mathbb{R},+)$ is a one-dimensional Lie group in the sense of Lie group. A one-parameter subgroup of a Lie group $G$ is a Lie-group homomorphism

$\gamma :(\mathbb{R},+)⟶G$

in the sense of Lie-group homomorphism, isomorphism, and automorphism. Thus $\gamma$ is smooth, is defined for every real parameter, and satisfies

$\gamma (s+t)=\gamma (s)\gamma (t),\gamma (0)=e,\gamma (-t)=\gamma (t{)}^{-1}\mathrm{.}$

The last two identities are consequences of the group-homomorphism law, not extra data. A smooth curve defined only on an interval around $0$ is therefore not yet a one-parameter subgroup, even if it satisfies the product law whenever all displayed parameters remain in that interval. The term also does not assert that the image is embedded or closed.

Both the domain and every Lie-group codomain are nonempty and boundaryless. For a zero-dimensional codomain the definition still permits, for example, the constant homomorphism; for a one-dimensional codomain it is unchanged. No metric, nondegeneracy, or finite endpoint occurs, and all maps and group operations are supplied explicitly, so no choice axiom is used. This is a definition, not a biconditional characterization.