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

Abstract

For a block $b$ of a normal subgroup of a finite group $G$, E. C. Dade has defined a normal subgroup $G\left[b\right]$ of the inertial group of $b$ in $G$. Let $S_G^0\left(b\right)$ 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 $S_G^0\left(b\right)/G\left[b\right]$ has a normal Sylow $p$-subgroup. Using this theorem, we show that (under the Alperin–McKay conjecture) the class-preserving outer automorphism group ${Out}_c\left(G\right)$ 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 $S_G^0\left(b\right)/G\left[b\right]$ and ${Out}_c\left(G\right)$ which are derived from the Alperin–McKay conjecture. Main results of the present paper depend on the classification theorem of finite simple groups.