Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] R. Carter and T. Hawkes, The $ \mathcal{F}$-normalizer of a finite soluble group, J. Algebra 5 (1967), 175-202. MR 34 #5914. MR 0206089 (34:5914)
  • [2] K. Doerk, Zur Theorie der Formationen endlicher auflösbarer Gruppen, J. Algebra 13 (1969), 345-373. MR 40 #237. MR 0246968 (40:237)
  • [3] G. Seitz and C. Wright, On complements of $ \mathcal{F}$-residuals in finite solvable groups, Arch. Math. (Basel) 21 (1970), 139-150. MR 42 #6117. MR 0271234 (42:6117)
  • [4] C. Wright, On screens and $ \mathcal{L}$-izers of finite solvable groups, Math. Z. 115 (1970), 273-282. MR 41 #6973. MR 0262365 (41:6973)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20D10

Retrieve articles in all journals with MSC: 20D10


Additional Information

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

American Mathematical Society