Grigory Ryabov — Page 3

arXiv2016

On Schur p-groups of odd order

A finite group $G$ is called a Schur group if any $S$-ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. We prove that the groups $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$, where $n\geq 1$, are Schur. Modulo previously obtained results, it follows that every noncyclic Schur $p$-group, where $p$ is an odd prime, is isomorphic to $\mathbb{Z}_3\times \mathbb{Z}_3 \times \mathbb{Z}_3$ or $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$, $n\geq 1$ .

math.GR↗

arXiv2015

On Schur 3-groups

Let $G$ be a finite group. If $Γ$ is a permutation group with $G_{right}\leqΓ\leq Sym(G)$ and $\mathcal{S}$ is the set of orbits of the stabilizer of the identity $e=e_{G}$ in $Γ$, then the $\mathbb{Z}$-submodule $\mathcal{A}(Γ,G)=Span_{\mathbb{Z}}\{\underline{X}:\ X\in\mathcal{S}\}$ of the group ring $\mathbb{Z} G$ is an $S$-ring as it was observed by Schur. Following Pöschel an $S$-ring $\mathcal{A}$ over $G$ is said to be schurian if there exists a suitable permutation group $Γ$ such that $\mathcal{A}=\mathcal{A}(Γ,G)$. 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.

math.GR↗