On the Shemetkov problem for Fitting classes

Wenbin Guo and Baojun Li

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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