Abstract
Denote by P the set of all primes and take a nonempty set π ⊆ P. A Fitting class F ≠ (1) is called normal in the class Sπ of all finite soluble π-groups or π-normal, whenever F ⊆ Sπ and for every G ∈ Sπ its F-injectors constitute a normal subgroup of G.
We study the properties of π-normal Fitting classes. Using Lockett operators, we prove a criterion for the π-normality of products of Fitting classes. A π-normal Fitting class is normal in the case π = P. The lattice of all solvable normal Fitting classes is a sublattice of the lattice of all solvable Fitting classes; but the question of modularity of the lattice of all solvable Fitting classes is open (see Question 14.47 in [1]). We obtain a positive answer to a similar question in the case of π-normal Fitting classes.
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Original Russian Text Copyright © 2015 Vorob’ev N.T. and Martsinkevich A.V.
Vitebsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 790–797, July–August, 2015; DOI: 10.17377/smzh.2015.56.406.
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Vorob’ev, N.T., Martsinkevich, A.V. Finite π-groups with normal injectors. Sib Math J 56, 624–630 (2015). https://doi.org/10.1134/S0037446615040060
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DOI: https://doi.org/10.1134/S0037446615040060