A remark on covering blocks (Q1078317)

Generalizing well known results of E. C. Dade and I. M. Isaacs the author shows in the main theorem of this note: Let N be a normal subgroup of G such that G/N is a p-group, and let b be a p-block of N. Then there is precisely one p-block B of G covering b. If a defect group D of B is abelian, then the following statements are equivalent: (1) There is a G- stable module in b, (2) b is G-stable, (3) \(G=DN\), (4) If \(\chi\in B\) is an irreducible ordinary or Brauer character of G, then \(\chi |_ N\) is irreducible, (5) Every ordinary irreducible character \(\psi\) of b has precisely \(q=| G:N|\) different extensions to G, all belonging to B. Every irreducible Brauer character \(\rho\) of b has precisely one extension \({\hat \rho}\) to G and \({\hat \rho}\in B.\) Furthermore, if these equivalent conditions are satisfied, then the decomposition matrix of B is obtained from the decomposition matrix of b by repeating each row q times, and for the Cartan matrices, one has \(C_ B=q^ Cb\).