Blocks of normal subgroups, automorphisms of groups, and the Alperin–McKay conjecture

For a block b of a normal subgroup of a finite group G, E. C. Dade has defined a normal subgroup G[b] of the inertial group of b in G. Let SG0(b) be the subgroup of G consisting of all elements of G fixing all irreducible characters of height 0 in b. Under the Alperin–McKay conjecture we show that SG0(b)/G[b] has a normal Sylow p-subgroup. Using this theorem, we show that (under the Alperin–McKay conjecture) the class-preserving outer automorphism group Outc(G) of a group G has p-length at most one for any prime p. This rectifies C. H. Sah’s incorrect proof that this group is solvable (under the Schreier conjecture). We obtain also other results on the structures of SG0(b)/G[b] and Outc(G) which are derived from the Alperin–McKay conjecture. Main results of the present paper depend on the classification theorem of finite simple groups.