Abstract:
Call a Fitting class $\mathfrak F$$\pi$-maximal if $\mathfrak F$ is (inclusion-)maximal in the class $\mathfrak C_\pi$ of all finite $\pi$-groups, where $\pi$ stands for a nonempty set of primes. We establish a $\pi$-maximality criterion for a Fitting class $\mathfrak F$ of finite $\pi$-groups: we prove that a nontrivial Fitting class $\mathfrak F$ is $\pi$-maximal if and only if there is a prime $p\in\pi$ such that, for every $\pi$-group $G$, the index of the $\mathfrak F$-radical $G_\mathfrak F$ in $G$ is equal to 1 or $p$. This implies Laue's familiar result on a necessary and sufficient condition of the maximality of an arbitrary Fitting class of finite groups in the class $\mathfrak C$ of all finite groups. The $\pi$-maximality criterion obtained also gives a confirmation of the negative solution of Skiba's Problem asking whether a local Fitting class has no inclusion-maximal Fitting subclasses (see Problem 13.50, The Kourovka Notebook: Unsolved Problems in Group Theory, 14th ed., Sobolev Institute of Mathematics, Novosibirsk, 1999).
Keywords:Fitting class, (inclusion-)maximal Fitting subclass, $\pi$-maximal Fitting class, $\pi$-maximality criterion for Fitting classes, class of all finite $\pi$-groups, local Fitting class, Lockett class.
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