On a question of T. Hawkes

In 1968 Hawkes proved that for every soluble finite group the intersection of the Fitting subgroup with a G -crucial maximal subgroup is its Fitting subgroup where G is a skeletal class of soluble groups. He suggested that this result might spring from a more general one. We prove that all hereditar...

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Bibliographic Details
Published in:Ricerche di matematica 2023-06, Vol.72 (1), p.443-451
Main Author: Murashka, Viachaslau I.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Geometry
Group theory
Mathematics
Mathematics and Statistics
Numerical Analysis
Probability Theory and Stochastic Processes
Subgroups
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Summary:In 1968 Hawkes proved that for every soluble finite group the intersection of the Fitting subgroup with a G -crucial maximal subgroup is its Fitting subgroup where G is a skeletal class of soluble groups. He suggested that this result might spring from a more general one. We prove that all hereditary Fitting classes F of soluble groups such that for every soluble finite group the intersection of the F -radical with an S -crucial maximal subgroup is its F -radical, where S is the least by inclusion skeletal class of soluble groups, are exactly classes of all σ -nilpotent soluble finite groups.
ISSN:0035-5038
1827-3491
DOI:10.1007/s11587-023-00767-z