Classification of abelian Schur groups I
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910199974985728 |
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| author | Ryabov, Grigory |
| author_facet | Ryabov, Grigory |
| contents | A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ 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 complete a classification of abelian Schur groups. In the present paper, we study schurity of several groups from the list. First, 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. Second, we establish nonschurity of some other groups from the list. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14215 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of abelian Schur groups I Ryabov, Grigory Group Theory Combinatorics 05E30, 20B25 A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ 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 complete a classification of abelian Schur groups. In the present paper, we study schurity of several groups from the list. First, 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. Second, we establish nonschurity of some other groups from the list. |
| title | Classification of abelian Schur groups I |
| topic | Group Theory Combinatorics 05E30, 20B25 |
| url | https://arxiv.org/abs/2602.14215 |