= Classification of nontrivial one-parameter subgroups
For $X\ne0$, the image of $t\mapsto e^{tX}$, with its intrinsic immersed <Lie subgroup> topology, is isomorphic to $(\mathbb R,+)$ or the <circle group>. The kernel is respectively zero or $\tau\mathbb Z$ for a least positive period $\tau$. The case $X=0$ is the trivial group. An irrational winding in a two-dimensional torus need not be an embedded or closed subgroup, so its intrinsic topology must be distinguished from the subspace topology.
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