On Schur 3-groups

Abstract

Let $G$ be a finite group. If $\Gamma$ is a permutation group with $G_{right}\le \Gamma \le Sym(G)$ and $\mathcal{S}$ is the set of orbits of the stabilizer of the identity $e=e_G$ in $\Gamma$, then the $\mathbb{Z}$-submodule $\mathcal{A}(\Gamma ,G)=Spa{n}_{\mathbb{Z}}\{\underline{X}:X\in \mathcal{S}\}$ of the group ring $\mathbb{Z}G$ is an $S$-ring as it was observed by Schur. Following Pöschel 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 $M_{3^n}=⟨a,b|a^{3^{n-1}}=b^3=e,a^b=a^{3^{n-2}+1}⟩$, where $n\ge 3$, are not Schur. Modulo previously obtained results, it follows that every Schur $p$-group is abelian whenever $p$ is an odd prime.

arXiv comment8 pages