On Schur $3$-groups

G. K. Ryabov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 223-231

Résumé

Let $G$ be a finite group. An $S$-ring $\mathcal{A}$ over $G$ is a subring of the group ring $\mathbb{Z}G$ that has a linear basis associated with a special partition of $G$. About 40 years ago R. Pöschel suggested the problem which can be formulated as follows: for which group $G$ every $S$-ring $\mathcal{A}$ over it is schurian, i.e. the partition of $G$ corresponding to $\mathcal{A}$ consists of the orbits of the one point stabilizer of a permutation group in $Sym(G)$ that contains a regular subgroup isomorphic to $G$. The main result of the paper says that such $G$ can not be non-abelian $p$-group, where $p$ is an odd prime. In fact, modulo known results, it was sufficient to show that for every $n\geq3$ there exists a non-schurian $S$-ring over the group $M_{3^n}=\langle a,b\;|\:a^{3^{n-1}}=b^3=e,a^b=a^{3^{n-2}+1}\rangle$.

Détail

Keywords: Permutation groups, Cayley schemes, $S$-rings, Schur groups.