On solvability of the first Hochschild cohomology of odd p-groups

We give a necessary and sufficient criterion for the solvability of ${\mathrm{HH}}^1(kP)$ as a Lie algebra, where $P$ is a $p$-group with $p$ odd, in terms of a directed graph constructed from the group $P$. This gives non-trivial results on the structure of such Lie algebras.