Classification of abelian Schur groups I

TL;DR

This work advances the classification of abelian Schur groups by proving that $E_4\times C_{p^k}$ and $E_4\times C_{pq}$ are Schur groups, while identifying several non-Schur families such as $C_{2p}\times C_{2^k}$ ($p$ odd, $k\ge3$) and $E_{16}\times C_p$ (for primes $p\ge5$). The authors develop a detailed structural theory for S-rings over products like $E_4\times C_n$, including a comprehensive description (Theorem e4cn) of when such rings are cyclotomic, tensor, or generalized wreath products, and use duality, tensor/star/wreath constructions, and minimality arguments to establish schurity in the favorable cases. The paper then closes the gaps for the main statements by constructing explicit nonschurian S-rings in the remaining families, thereby delineating exactly which abelian groups from the list are Schur and which are not. Overall, the results solidify the landscape of abelian Schur groups and provide practical machinery for verifying schurity via S-ring decompositions and automorphism-structure analysis.

Abstract

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations of $G$. The list of all possible abelian Schur groups was obtained by Evdokimov, Kovács, and Ponomarenko in 2016. In two papers, we finish a classification of abelian Schur groups. In the present paper, we study schurity of several groups from the list. Namely, we prove that a direct product of the elementary abelian group of order 4 and a cyclic group, whose order is an odd prime power or a product of two distinct odd primes, is a Schur group and establish nonschurity of some other groups from the list.

Theorems & Definitions (94)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof