  Let $G$ be a finite group. If $\Gamma$ is a permutation group with
$G_{right}\leq\Gamma\leq Sym(G)$ and $\mathcal{S}$ is the set of orbits of the
stabilizer of the identity $e=e_{G}$ in $\Gamma$, then the
$\mathbb{Z}$-submodule
$\mathcal{A}(\Gamma,G)=Span_{\mathbb{Z}}\{\underline{X}:\ X\in\mathcal{S}\}$ of
the group ring $\mathbb{Z} G$ is an $S$-ring as it was observed by Schur.
Following P\"{o}schel an $S$-ring $\mathcal{A}$ over $G$ is said to be schurian
if there exists a suitable permutation group $\Gamma$ such that
$\mathcal{A}=\mathcal{A}(\Gamma,G)$. A finite group $G$ is called a Schur group
if every $S$-ring over $G$ is schurian. We prove that the groups
$M_{3^n}=\langle a,b\;|\:a^{3^{n-1}}=b^3=e,a^b=a^{3^{n-2}+1}\rangle$, where
$n\geq3$, are not Schur. Modulo previously obtained results, it follows that
every Schur $p$-group is abelian whenever $p$ is an odd prime.
