The $D_\pi$-Property in a Class of Finite Groups
Algebra i logika, Tome 41 (2002) no. 3, pp. 335-370
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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.
Mots-clés :
$D_\pi$-group, sporadic group
Keywords: alternating group, simple group of Lie type, Hall subgroup.
Keywords: alternating group, simple group of Lie type, Hall subgroup.
@article{AL_2002_41_3_a4,
author = {D. O. Revin},
title = {The $D_\pi${-Property} in {a~Class} of {Finite} {Groups}},
journal = {Algebra i logika},
pages = {335--370},
publisher = {mathdoc},
volume = {41},
number = {3},
year = {2002},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AL_2002_41_3_a4/}
}
D. O. Revin. The $D_\pi$-Property in a Class of Finite Groups. Algebra i logika, Tome 41 (2002) no. 3, pp. 335-370. http://geodesic.mathdoc.fr/item/AL_2002_41_3_a4/