  This paper studies the canonical symmetric connection $\nabla$ associated to
any Lie group $G$. The salient properties of $\nabla$ are stated and proved.
The Lie symmetries of the geodesic system of a general linear connection are
formulated. The results are then applied to $\nabla$ in the special case where
the Lie algebra $\g$ of $G$, has a codimension one abelian nilradical. The
conditions that determine a Lie symmetry in such a case are completely
integrated. Finally the results obtained are compared with some
four-dimensional Lie groups whose Lie algebras have three-dimensional abelian
nilradicals, for which the calculations were performed by MAPLE.
