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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On integrated screens


Author: J. C. Beidleman
Journal: Proc. Amer. Math. Soc. 42 (1974), 36-38
MSC: Primary 20D10
DOI: https://doi.org/10.1090/S0002-9939-1974-0327891-1
MathSciNet review: 0327891
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0327891-1
Keywords: Screen, formation, $ \mathcal{L}$-izer, solvable
Article copyright: © Copyright 1974 American Mathematical Society