On Intersections of Abelian and Nilpotent Subgroups in Finite Groups. II

Mathematical Notes - Let G be a finite group, and let A and B be, respectively, an Abelian and a nilpotent subgroup in G. In the present paper, we complete the proof of the theorem claiming that...     

On intersections of abelian and nilpotent subgroups in finite groups. I

We consider Pontryagin’s generalized nonstationary example with identical dynamic and inertial capabilities of the players under phase constraints on the evader’s states. The boundary of the phase constraints is not a “death line” for the evader. The set of admissible controls is a ball centered at the origin, and the terminal sets are the origin. We obtain sufficient conditions for a multiple capture of one evader by a group of pursuers in the case when some functions corresponding to the initial data and to the parameters of the game are recurrent.     

Intersections of two nilpotent subgroups in finite groups with socle <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Given a finite group G with socle isomorphic to L ₂(q), q ≥ 4, we describe, up to conjugacy, all pairs of nilpotent subgroups A and B of G such that A ∩ B ^g ≠ 1 for all g ∈ G.     

Finite groups in which the normalizers of pairwise intersections of Sylow 2-subgroups have odd indices

All finite semisimple groups are described in which the normalizers of pairwise intersections of Sylow 2-subgroups have odd indices; thereby, Problem 5.14(v) from the Kourovka Notebook is solved in the main.     

Generation of finite groups by the class of conjugate Abelian subgroups

Using the basic theorem on the classification of finite simple groups, we answer one of the questions concerning the generation of finite groups by the class of conjugate Abelian subgroups. Supported by RFFR grant No. 93-01-01529. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 288–293, May–June, 1996.     

Bismuth-induced enhancement of magneto-optical effects in iron garnets: A theoretical analysis

A semiquantitative model of circular magneto-optical effects in iron garnets is constructed within the concept of charge-transfer transitions and the existing qualitative notions. In the framework of the proposed model, the drastic enhancement of circular magneto-optical effects in R₃Fe₅O₁₂ iron garnets containing impurities of Bi³⁺ or Pb²⁺ ions is explained by the increase in the oxygen contribution to the spin-orbit coupling constant of the (FeO₆)^9? and (FeO₄)^5? complexes (the main magneto-optically active centers in iron garnets). This increase is associated with the covalent admixture of the Bi³⁺ (or Pb²⁺) 6p orbitals (with a giant one-electron spin-orbit coupling constant) to the oxygen 2p orbitals. The influence of the substitution does not reduce to an enhancement of the oxygen spin-orbit interaction alone but also leads to the appearance of the effective anisotropic tensor contributions to the spin-orbit interaction and circular magneto-optical effects. These contributions to the magneto-optical effects in garnets are estimated. The influence of an inhomogeneous bismuth distribution on the magneto-optical effects in Y_3?xBi_xFe₅O₁₂ garnets is investigated using computer simulation. Analysis of the available experimental data on the magneto-optical effects in garnets confirms the validity of the theoretical model proposed.     

Phase separation and manifestation of nanoscopic nonuniformities in the optical spectra of manganites

The features observed in the frequency and temperature dependences of the optical spectra of doped manganites can be accounted for by assuming the latter to have inhomogeneous phase composition. The spectral response of the optical conductivity of the La_1?xSr_xMnO₃ system was calculated in the effective-medium approximation over a broad range of x variation.     

An Analog for the Frattini Factorization of Finite Groups

Using the classification of finite simple groups, we prove that if H is an insoluble normal subgroup of a finite group G, then H contains a maximal soluble subgroup S such that G=HN_G(S). Thereby Problem 14.62 in the Kourovka Notebook is given a positive solution. As a consequence, it is proved that in every finite group, there exists a subgroup that is simultaneously a -projector and a -injector in the class, , of all soluble groups.