On Huppert’s condition $B$

Abstract

Let $\mathcal {F}$ be a saturated formation of finite soluble groups. $\mathcal {F}$ is said to satisfy condition $\text {B}$ if and only if (a) $\mathcal {F}$ is subgroup-closed, and (b) $G \in \mathcal {F}$ and $N$ a minimal normal subgroup of $G$ implies $\operatorname {Aut} (N) \in \mathfrak {F}$. The purpose of this note is to characterize those saturated formations of finite soluble groups which satisfy condition ${\mathbf {B}}$.