The projective cover of the trivial module over a group algebra of a finite group. (English) Zbl 1304.20015
Let \(k\) be a field of characteristic \(p>0\), and let \(P\) be the projective cover of the trivial \(kG\)-module where \(G\) is a finite group. The author shows that \(P\) has Loewy length 4 if and only if \(p=2\) and \(O^{2'}(G/O_{2'}(G))\) is isomorphic to \(C_4\), \(C_2\times C_2\times C_2\), \(C_2\times\text{PSL}_2(q)\) for some \(q\equiv 3\pmod 4\), or \(\text{PGL}_2(q)\) for some \(q\equiv 3\pmod 4\). It follows that the principal 2-block of \(kG\) has Loewy length 4 if and only if \(O^{2'}(G/O_{2'}(G))\) is isomorphic to \(C_4\), \(C_2\times C_2\times C_2\), \(C_2\times\text{PSL}_2(q)\) for some \(q\equiv 3\pmod 8\), or \(\text{PGL}_2(q)\) for some \(q\equiv 3\pmod 8\).
Reviewer: Burkhard Külshammer (Jena)
MSC:
| 20C20 | Modular representations and characters |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
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