On subsections of blocks and Brauer pairs (Q1591247)

Let \(p\) be prime, let \(G\) be a finite group, and \(H\) a subgroup of \(G\). Let \(B\) and \(b\) be \(p\)-blocks of \(G\) and \(H\), respectively. Let \(\sigma(B,b)=|G:H|\zeta_b(1)/\zeta(1)\) for an irreducible character \(\zeta\) in \(B\), where \(\zeta_b\) is the \(b\)-component of \(\zeta_H\). Assume \(d(B)\geq d(b)\). Then \(B\) and \(b\) are linked and have a common defect group if and only if \(\sigma(B,b)\) is nonzero modulo~\(p\). The author also studies \(\sigma(B,b)\) in the case where \(b^G=B\) and \(d(b)=d(B)\). If \(P\) is a \(p\)-subgroup of \(G\) and \(b_P\) is a block of \(PC_G(P)\) with defect group \(P\), \((P,b_P)\) is called a Brauer pair in \(G\). Conditions are given for \(b_P\) and \(b_Q\) to be linked, where \((P,b_P)\) and \((Q,b_Q)\) are Brauer pairs with \(Q\) a normal subgroup of \(P\), slightly improving a result of \textit{R.~Brauer} [Am. J. Math. 89, 1115-1136 (1967; Zbl 0174.05401)].