On finite groups all of whose \(p\)-blocks are of the highest defect
Let \(p\) be a prime integer. By definition, if \(G\) is a finite group, then \(G\) is full \(p\)-defective if every \(p\)-block of \(G\) is of the highest defect. The main result of this paper is: Theorem 4.3. Let \(G\) be an arbitrary finite group and let \(P\) be a Sylow \(p\)-subgroup of \(G\) and set \(M=O_ p(C_ G(P))\). Then \(G\) is full \(p\)-defective if and only if the following two conditions hold: (1) \(O_{p'}(G)=\cup_{x\in G}M^ x\); and (2) \(F^*(G/O_{p'}(G))=(O_ p(G)O_{p'}(G))/O_{p'}(G)\) if \(p\) is odd and \(F^*(G/O_{2'}(G))=(F\times O_{2'}(G))/O_{2'}(G)\) where \(F=L(G)O_ 2(G)\) and all components of \(L(G)\) are of type \(M_{22}\) or \(M_{24}\) if \(p=2\). The proof of this result begins, in Section 3, by examining the list of finite simple groups and demonstrating that \(M_{22}\) and \(M_{24}\) with \(p=2\) are the only simple groups \(G\) that are full \(p\)-defective with \(p\mid|G|\). Section 4 presents a proof of Theorem 4.3 and Section 5 presents some applications.
scientific article; zbMATH DE number 4063352
scientific article