The Alperin argument revisited (Q1355284)

Let \(p\) be a prime, let \(G\) be a finite \(p\)-solvable group and let \(B\) be a \(p\)-block of \(G\). Inspired by Alperin's argument proving the Glauberman correspondence in case that a \(p\)-group acts on a \(p'\)-group via Brauer's first main theorem the author investigates \(p\)-blocks \(B\) of \(G\) which contain only one Brauer character. To state the main results we need the definition of a uniqueness triple. This is a triple \((U,\gamma,\theta)\) where \(\gamma\in\text{IBr}_p(U)\), \(\theta\in\text{Irr}(O_{p'}(U))\) and for a \(p'\)-Hall subgroup \(K\) of \(U\), the restriction \(\gamma_K\) is irreducible and fully ramified over \(\theta\). With this notation the author proves in Theorem 5.4: \(\text{IBr}_p(G)=\{\varphi\}\) if and only if \(\varphi=\gamma^G\) where \((U,\gamma,\theta)\) is a uniqueness triple. Suppose this is the case, then induction of characters defines a bijection between \(\text{Irr}(U|\theta)\) and \(\text{Irr}(B)\). Even more can be said: Let \(Q\in\text{Syl}_p(U)\) and let \(\theta^*\) denote the Glauberman correspondent of \(\theta\) in \(C_{O_{p'}(U)}(Q)\). Then with the notation as above the main theorem says: a) \(\theta^*\) is fully ramified with respect to \(N_K(Q)\). Furthermore the unique irreducible constituent \(\delta^*\) of \(\theta^{*N_K(Q)}\) extends to a unique \(\gamma^*\in\text{IBr}_p(N_U(Q))\) and \(\gamma^{*N_G(Q)}\in\text{IBr}_p(N_G(Q))\); b) Assume that \(\varphi=\mu^G\) for some \(\mu\in\text{IBr}_p(V)\). Moreover, assume that \(H\) is a \(p'\)-Hall subgroup of \(V\) and \(\mu_H\) is fully ramified with respect to \(O_{p'}(V)\). Then \(\gamma^{*N_G(Q)}=\mu^{*N_G(Q)}\) for \(Q\in\text{Syl}_p(V)\). The results are stated more generally, namely for \(\pi\)-separable groups and \(\pi\)-partial characters as introduced by Isaacs. The formulation in this review is only for \(\pi=p'\), in which the set of \(\pi\)-partial characters of \(G\) coincides with \(\text{IBr}_p(G)\).