On integrated screens

Abstract:

Let $\mathcal {L}$ be a screen with support $\pi$ and let $\mathcal {F}$ denote the saturated formation of finite solvable groups which is locally induced by $\mathcal {L}$. For each prime p, let $\mathcal {M}(p) = \mathcal {L}(p) \cap \mathcal {F}$. Then $\mathcal {M}$ is an integrated screen which locally induces $\mathcal {F}$ and $\mathcal {M} \subseteq \mathcal {L}$. The purpose of this note is to prove the following theorems. Theorem 1. Assume that for each finite solvable group G the $\mathcal {L}$-izers of G satisfy the strict cover-avoidance property. Then $\mathcal {L}$ is an integrated screen; that is $\mathcal {L}(p) \subseteq \mathcal {F}$ for each prime p. Theorem 2. Assume that for each group G an $\mathcal {M}$-izer of an $\mathcal {L}$-izer of G is an $\mathcal {M}$-izer of G. Then $\mathcal {L}(p) = \mathcal {M}(p)$ for each prime p.