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 saturated formations which are special relative to the strong covering-avoidance property


Authors: J. C. Beidleman and A. R. Makan
Journal: Proc. Amer. Math. Soc. 47 (1975), 29-36
DOI: https://doi.org/10.1090/S0002-9939-1975-0357589-6
MathSciNet review: 0357589
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: Let $ \mathfrak{F}$ be a saturated formation of finite soluble groups. Let $ G$ be a finite soluble group and $ F$ an $ \mathfrak{F}$-projector of $ G$. Then $ F$ is said to satisfy the strong covering-avoidance property if (i) $ F$ either covers or avoids each chief factor of $ G$, anaaa $ F \cap L/F \cap K$ is a chief factor of $ F$ whenever $ L/K$ is a chief factor of $ G$ coverzd by $ F$Let $ {\mathcal{C}_\mathfrak{F}}$ denote the class of all finite soluble $ G$ in which the $ \mathfrak{F}$-projectors satisfy the strong covering-avoidance property. $ {\mathcal{C}_\mathfrak{F}}$ is a formation. Let $ {\mathcal{Y}_\mathfrak{F}}$ be the class of groups $ G$ in which an $ \mathfrak{F}$-normalizer is also an $ \mathfrak{F}$-projector. $ {\mathcal{Y}_\mathfrak{F}}$aa is a formation studied by Klaus Doerk. Note that $ {\mathcal{Y}_\mathfrak{F}} \subseteq {\mathcal{C}_\mathfrak{F}}$. $ \mathfrak{F}$ is said to be $ \mathcal{C}$-special if $ {\mathcal{C}_\mathfrak{F}} = {\mathcal{Y}_\mathfrak{F}}$. he puraaaaaaaaaaas note is to sdy $ \mathcal{C}$-special formations. Two characterizations of $ \mathcal{C}$-special formations are given. Let $ i$ be a positive integer and let $ {\mathfrak{N}^{(i)}}$ denote the class of finite soluble groups $ G$ whose Fitting length is at most $ i$. Then $ {\mathfrak{N}^{(i)}}$ is $ \mathcal{C}$-special. Finally, the formation $ {\mathcal{C}_\mathfrak{F}}$aaaaaturated if and only if $ \mathfrak{F}$ is the class of all finite soluble groups.


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

  • [1] R. W. Carter and T. O. Hawkes, The $ \mathfrak{F}$-normalizers 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 auflösbaren Gruppen, J. Algebra 13 (1969), 345-373. MR 40 # 237. MR 0246968 (40:237)
  • [3] K. Doerk and T. Hawkes, Two questions in the theory of formations, J. Algebra 16 (1970), 456-460. MR 42 # 7775. MR 0272894 (42:7775)
  • [4] W. Gaschütz, Zur Theorie der endlichen auflösbaren Gruppen, Math. Z. 80 (1962/63), 300-305. MR 31 # 3505. MR 0179257 (31:3505)
  • [5] T. O. Hawkes, An example in the theory of soluble groups, Proc. Cambridge Philos. Soc 67 (1970), 13-16. MR 40 # 1477. MR 0248225 (40:1477)
  • [6] B. Huppert, Endliche Gruppen I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin and New York, 1967. MR 37 # 302. MR 0224703 (37:302)
  • [7] -, Zur Theorie der Formationen, Arch. Math. (Basel) 19 (1968), 561-574 (1969). MR 39 # 5697. MR 0244382 (39:5697)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0357589-6
Keywords: Saturated formation, soluble, projector, normalizer
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society