Simple modules for groups with Abelian Sylow 2-subgroups are algebraic. (English) Zbl 1173.20001
A module is called algebraic if it satisfies a polynomial with integer coefficients with direct sum and tensor product as addition and multiplication. The paper studies the following conjecture: For a 2-block \(B\) with Abelian defect group all simple \(B\)-modules are algebraic. In a paper of the author et al. (to appear) it is proved that if \(B\) is a block with the Klein 4-group as defect group then all simple \(B\)-modules are algebraic.
The main result of the present paper is that if \(G\) is a group with Abelian 2-Sylow subgroups and \(K\) is a field of characteristic 2 then all simple \(KG\)-modules are algebraic. A closely related theorem is that of T. R. Berger [Proc. Conf. Finite Groups, Park City 1975, 541-553 (1976; Zbl 0375.20004)]: if \(G\) is a soluble group and \(K\) is an algebraically closed field of prime characteristic then all \(KG\)-modules are algebraic.
The main tools used by the author in the proof is the characterization of finite groups with Abelian 2-Sylow subgroups by J. H. Walter [Ann. Math. (2) 89, 405-514 (1969; Zbl 0184.04605)], and the computer algebra software Magma used for determining simple modules for the Janko-group \(J_1\).
The main result of the present paper is that if \(G\) is a group with Abelian 2-Sylow subgroups and \(K\) is a field of characteristic 2 then all simple \(KG\)-modules are algebraic. A closely related theorem is that of T. R. Berger [Proc. Conf. Finite Groups, Park City 1975, 541-553 (1976; Zbl 0375.20004)]: if \(G\) is a soluble group and \(K\) is an algebraically closed field of prime characteristic then all \(KG\)-modules are algebraic.
The main tools used by the author in the proof is the characterization of finite groups with Abelian 2-Sylow subgroups by J. H. Walter [Ann. Math. (2) 89, 405-514 (1969; Zbl 0184.04605)], and the computer algebra software Magma used for determining simple modules for the Janko-group \(J_1\).
Reviewer: János Kurdics (Nyíregyháza)
MSC:
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 20C20 | Modular representations and characters |
| 20D05 | Finite simple groups and their classification |
Software:
MagmaReferences:
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