On \(2\)-blocks of characters with defect \(1\) (Q1921908)

Let \(G\) be a finite group. We denote the order of \(G\) by \(2^a\cdot u\) for an odd number \(u\). The article under review discusses ordinary irreducible complex characters \(\chi\) of \(G\) with degree \(2^{a-1}\cdot d\) for an odd number \(d\). Those characters are said to be of defect 1. Then, there is a unique irreducible complex character \(\chi_0\) of \(G\) such that \(\chi+\chi_0\) is a projective 2-modular character, and there is a 2-block \(B\) of \(G\) such that this is the only projective character in \(B\). The Green correspondents \(\zeta\) and \(\zeta_0\) of the characters \(\chi\) and \(\chi_0\) in the normalizer of their common cyclic defect group \(D\) behave differently. The character, say \(\chi\) whose Green correspondent has the defect group in the kernel is called non exceptional, the other is called exceptional. The author proves that if there is a finite group \(\widehat G\) in which \(G\) is normal of index a power of 2 and \(\widehat G\) operates as automorphisms of \(B\), then there is a 2-rational character of \(\widehat G\) extending \(\chi\). For \(\chi_0\) this cannot be true. But, if there is a finite group \(\widehat G\) in which \(G\) is normal of index a power of 2 and \(\widehat G\) operates as automorphisms of \(B\), then there is a unique 2-block \(\widehat B\) of \(\widehat G\) covering \(B\) of defect group \(\widehat D\) containing \(D\). Then, \(\overline G=\widehat D/D\) is a 2-rational representation group for \(\chi_0\) with regard to \(\widehat G\).