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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
MathSciNet review: 0357589
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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.

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Keywords: Saturated formation, soluble, projector, normalizer
Article copyright: © Copyright 1975 American Mathematical Society