  In this paper, it is shown that (with no additional assumptions) on a compact
7-dimensional manifold which admits a $G_2$-structure soliton solutions to the
Laplacian flow of R. Bryant can only be shrinking or steady. We also show that
the space of symmetries (vector fields that annihilate via the Lie derivative)
of a torsion-free $G_2$-structure on a compact 7-manifold is canonically
isomorphic to $H^1(M,\mathbb{R})$. Some comparisons with Ricci solitons are
also discussed, along with some future directions of exploration.
