The Alperin argument revisited

Abstract

It was observed by J.L. Alperin ([1]) that the Glauberman Correspondence (in the case a /?-group acts on a //-group) was a consequence of the Brauer First Main Theorem. Namely, if a /?-group P acts on a //-group G and χ is an irreducible P-invariant character of G, then χ uniquely extends to some Brauer character φ of Γ = GP. This Brauer character lies in a /?-block B of defect P and, in fact, φ is the only modular character in B. If b is the /7-block of NΓ(P) = CG(P) x P with b = B, then 6 contains a unique Brauer character φ* and χ* = φcG(P) is irreducible. This is the Glauberman Correspondence via the Alperin Argument. (Perhaps, this is a good place to stress that, although the /7-group case is certainly important, the Glauberman Correspondence is defined for general P solvable). Later, H. Nagao extended the Glauberman Correspondence (also in the /7-group case) to a noncoprime situation. If G is a normal subgroup of Γ with /?-power index and χ is a Γ-invariant /7-defect zero character of G, then χ is naturally associated with an irreducible /7-defect zero character of CG(P), where now P is some /^-subgroup of Γ complementing G. Notice that Nagao's map is, again, another application of the Alperin Argument: since χ is a Brauer character and Γ/G is a/?-group, then there is a unique Brauer character φ of Γ over χ (Green's Theorem); the block B of φ has a unique modular character (because B covers the block {χ})9 and the defect group P of B complements G in Γ (Fong's Theorem). Now the /7-block b of NΓ(P) = CG(P)x P with b G = B, has a unique modular character φ* with χ* = ψcG(P) irreducible. (The Nagao correspondence in the non /?-group case was constructed in [11].) As we see, there is an essential idea above: find blocks B with only one Brauer character and prove that Brauer First Main correspondents b satisfy the same property. If this is the case, the existence of a natural map, has been shown. In the Glauberman-Nagao conditions, this is not a problem: since NΓ(P) = PCG(P), it follows that every block of defect P of NΓ(/>) has a unique modular character. To prove this fact in general, however, seems deep and it is a consequence