Principal blocks with six ordinary irreducible characters (Q6565659)

Let \(G\) be a finite group and \(p \in \pi(G)\). The principal \(p\)-block of \(G\) is the one containing the principal character \(\mathbf{1}_{G}\) and that its defect groups are the Sylow \(p\)-subgroups of \(G\). Let \(B_{0}\) denote the principal \(p\)-block of \(G\), and let \(k(B_{0})\) denote the number of ordinary irreducible characters of \(B_{0}\).\N\NThe purpose of the paper under review is to determine the structure of a Sylow \(p\)-subgroups \(P\) of a finite group whose principal \(p\)-blocks have precisely six ordinary irreducible characters. In this case, the authors prove that \(|P|=9\). In the main theorem, they also determine in detail the structure of these groups.\N\NTheorem B: Let \(G\) be a finite group, \(p\) a prime, and \(P\in \mathrm{Syl}_{p}(G)\). Let \(B_{0}\) denote the principal \(p\)-block of \(G\). Then \(k(B_{0})=6\) if and only if precisely one of the following holds: (i) \(P=\mathsf{C}_{9}\) and \(|N_{G}(P):C_{G}(P)|= 2\). (ii) \(P=\mathsf{C}_{3} \times \mathsf{C}_{3}\) and either \(N_{G}(P)/C_{G}(P) \in \{\mathsf{C}_{4}, \mathsf{Q}_{8}\}\) or \(N_{G}(P)/C_{G} (P) \simeq \mathsf{C}_{2}\) acts fixed-point freely on \(P\).