On abelian Schur 3-groups

Grigory Ryabov
Novosibirsk State University



Content: 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 $e$ 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 \emph{schurian} if there exists a suitable permutation group $\Gamma$ such that $\mathcal{A}=\mathcal{A}(\Gamma,G)$. A finite group $G$ is called a \emph{Schur group} if every $S$-ring over $G$ is schurian. We prove that the groups $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$, where $n\geq1$, are Schur. Modulo previously obtained results, it follows that every non-cyclic Schur $p$-group is isomorphic to $\mathbb{Z}_3\times \mathbb{Z}_3\times \mathbb{Z}_3$ or $\mathbb{Z}_3\times \mathbb{Z}_{3^n},~n\geq 1,$ whenever $p$ is an odd prime.

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