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On schurity of finite abelian groups

Authors
  • Evdokimov, Sergei
  • Kovács, István
  • Ponomarenko, Ilya
Type
Published Article
Publication Date
Sep 03, 2013
Submission Date
Sep 03, 2013
Identifiers
arXiv ID: 1309.0989
Source
arXiv
License
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External links

Abstract

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3\times Z_{3^k}$ or $Z_3\times Z_3\times Z_p$ where $k\ge 1$ and $p$ is a prime. In addition, we prove that $Z_2\times Z_2\times Z_p$ is a Schur group for every prime $p$.

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