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This article is cited in 26 scientific papers (total in 26 papers)
The $D_\pi$-Property in a Class of Finite Groups
D. O. Revin The Specialized Educational Scientific Center of Novosibirsk State University
Abstract:
A finite group $G$ is a $D_\pi$-group for some set $\pi$ of primes if maximal $\pi$-subgroups of $G$ are all conjugate. Assume that every non-Abelian composition factor of the $D_\pi$-group $G$ is isomorphic either to an alternating group, or to one of the sporadic groups, or to a simple group of Lie type over a field whose characteristic belongs to $\pi$. We prove that an extension of $G$ by an arbitrary $D_\pi$-group and every normal subgroup of $G$ are $D_\pi$-groups. This gives partial answers to Questions 3.62 and 13.33 in the “Kourovka Notebook”. Also, we describe all $D_\pi$-groups whose composition factors are isomorphic to alternating, sporadic, and Lie-type groups whose characteristics belong to $\pi$. And bring to a close the description of Hall subgroups in sporadic groups, initiated by F. Gross.
Keywords:
$D_\pi$-group, alternating group, sporadic group, simple group of Lie type, Hall subgroup.
Received: 16.08.2000 Revised: 16.08.2001
Citation:
D. O. Revin, “The $D_\pi$-Property in a Class of Finite Groups”, Algebra Logika, 41:3 (2002), 335–370; Algebra and Logic, 41:3 (2002), 187–206
Linking options:
https://www.mathnet.ru/eng/al187 https://www.mathnet.ru/eng/al/v41/i3/p335
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