A new character correspondence in groups of odd order.
Let \(p\) be a prime and let \(G\) be a finite \(p\)-solvable group. In a previous paper, [Trans. Am. Math. Soc. 354, No. 7, 2759-2773 (2002; Zbl 0998.20012)], the author showed that for each irreducible complex character \(\chi\) of \(G\), there is a pair \((Q,\delta)\), where \(Q\) is a \(p\)-subgroup of \(G\) and \(\delta\) is an irreducible character of \(Q\). The pair is uniquely determined up to \(G\)-conjugacy and is called a vertex of \(\chi\). Let \(\text{Irr}(G| Q,\delta)\) denote the set of irreducible characters of \(G\) with vertex \((Q,\delta)\). Let \(\text{Irr}_0(G)\) denote the set of irreducible characters of \(G\) that have defect 0 modulo \(p\) and let \(N_G(Q,\delta)\) be the stabilizer subgroup of \(\delta\) in \(N_G(Q)\). The author's main result is the following. Suppose that \(G\) has odd order and let \(Q\) be a \(p\)-subgroup of \(G\) and \(\delta\) be an irreducible character of \(Q\). Then there is a natural injection \[ \text{Irr}(G| Q, \delta)\to\text{Irr}_0(N_G(Q,\delta)/Q). \] The author notes that when \(\delta=1_Q\) is the principal character of \(Q\), \(\text{Irr}(G| Q,\delta)\) is then a canonical set of liftings of the irreducible Brauer characters of \(G\) with traditional vertex \(Q\), and that his theorem provides a canonical bijection between the irreducible Brauer characters and the Alperin weights for a group of odd order (a result already proved by him [in Math. Z. 212, No. 4, 535-544 (1993; Zbl 0795.20005)]). He also obtains further applications of his injection, including a stronger form of the McKay conjecture for groups of odd order, as originally proved by I. M. Isaacs.
scientific article; zbMATH DE number 1995419
scientific article