  Our main is to study periodic orbits of linear and invariant flows on a real,
connected Lie group. Since each linear flow $\varphi_t$ has a derivation
associated $\mathcal{D}$, we show that the existence of periodic orbits of
$\varphi_t$ is based on the eigenvalues of the derivation $\mathcal{D}$. From
this, we study periodic orbits of a linear flow on noncompact, semisimple Lie
groups, and we work with periodic orbits of a linear flow on connected, simply
connected, solvable Lie groups of dimension 2 or 3.
