Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 48, No. 1, pp. 281-289 (2007)

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On the Shemetkov problem for Fitting classes

Wenbin Guo and Baojun Li

Department of Mathematics, Xuzhou Normal University, Xuzhou,221116, P.R.China, e-mail:; Department of Mathematics, University of Science and Technology of China, Hefei 230026, P.R. China

Abstract: Suppose that $\pi$ be a set of primes and ${\frak F}$ a local Fitting class. Let $K_{\pi}({\frak F})$ be the set of finite $\pi$-soluble groups with a Hall $\pi$-subgroup belonging to ${\frak F}$. In this paper, we show that the class $K_{\pi}({\frak F})$ is a local Fitting class. Thus, an interesting Shemetkov question for Fitting classes will be answered positively. By using the result, the ${\frak F}$-radical of a Hall $\pi$-subgroup of a finite $\pi$-soluble group is described. For a $H$-function $f$, we also give the definition and its description of $f$-radical of a finite $\pi$-soluble group. Some known important results follow.

Keywords: Fitting class; local Fitting class; Hall $\pi$-subgroup; ${\frak F}$-radical

Classification (MSC2000): 20D10, 20D20, 20D25

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Electronic version published on: 14 May 2007. This page was last modified: 27 Jan 2010.

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