[논문 리뷰] Classification of abelian Schur groups I

Grigory Ryabov|arXiv (Cornell University)|2026. 02. 15.

Finite Group Theory Research인용 수 0

한 줄 요약

논문은 E4 × Cp^k 및 E4 × Cpq가 Schur 그룹임을 보이고, 다른 목록에 있는 그룹들 중 비-Schur 사례를 식별함으로써 가환 Schur 그룹 분류를 진행한다.

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.

연구 동기 및 목표

• Finish the classification of abelian Schur groups by examining the remaining families.
• Determine schurity of specific groups from the known list of abelian Schur groups.
• Show that certain products of elementary abelian and cyclic groups are Schur groups.
• Identify and construct examples of nonschurian S-rings for particular groups from the list.

제안 방법

• Utilize the theory of S-rings, their isomorphisms, and schurity criteria.
• Descriptively classify S-rings over the groups E4 × Cp^k and E4 × Cpq.
• Apply multiplier theorems and structure results for abelian groups to control basic sets.
• Employ tensor, generalized wreath, and star products to build or decompose S-rings.
• Analyze dual S-rings and their relation to automorphism groups to verify schurity.
• Construct nonschurian S-rings as generalized wreath products to demonstrate non-schurity in certain cases.]
• research_questions([

실험 결과

연구 질문

• RQ1 Which abelian groups are Schur groups within the known nine families from prior work?
• RQ2 Are the groups E4 × Cp^k and E4 × Cpq Schur groups for odd primes p, q and k ≥ 1?
• RQ3 Do there exist non-Schur instances among the listed non-cyclic, non-elementary abelian families (e.g., C2p × C2^k, C2p × C2^k with k ≥ 3, E16 × Cp for certain p)?
• RQ4 How do generalized wreath, tensor, and star product constructions influence schurity for the groups in the list?

주요 결과

• The groups E4 × Cp^k and E4 × Cpq (p, q odd primes; k ≥ 1) are Schur groups.
• Groups C2p × C2^k with p odd and k ≥ 3 are not Schur groups.
• The group E16 × Cp is Schur if and only if p = 3.
• New nonschurian S-rings over C2p × C8 and over E16 × Cp are constructed via generalized wreath products.
• Some Schur groups from the nine families are confirmed, while others are shown to be nonschurian in specific instances.
• The paper characterizes S-rings over the studied groups, enabling verification of schurity via automorphism groups and product constructions.