Finite groups

The present survey is formed primarily on the basis of works reviewed in the Referativnyi Zhurnal Matematika during 1976–1983 and is a continuation of surveys of the same name published in 1966, 1971, 1976 in the series Albegra, Topologiya, Geometriya (INT). Principal attention is devoted to finite simple groups and their classification.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 24, pp. 3–120, 1986.     

On automorphisms of a generalized octagon of order (2, 4)

Possible orders and fixed-point subgraphs of automorphisms of a generalized octagon of order (2, 4) are found. The vertex-symmetric generalized octagon of order (2, 4) is proved to be classical.     

Metal-insulator transition in double cobaltites RBaCo2O5.5 (R = Eu, Gd): Specific features of their optical properties

The temperature dependence of the optical properties of GdBaCo₂O_5.5 in the region of the metal-insulator transition was studied in the spectral range λ = 4.0–0.258 μm (E = 0.3–4.8 eV) by the spectroscopic ellipsometry method. The electronic structure of the crystal at T = 300 K was calculated by the linearized muffin-tin orbital method in the local-density approximation with inclusion of on-site Coulomb interaction (LDA + U). The general features of the behavior of the optical properties of GdBaCo₂O_5.5 and EuBaCo₂O_5.5 are discussed. It is shown that the optical response on the metal side of the metal-insulator transition is determined by the redistribution of the optical-conductivity spectral weight in going from high to low energies and is significantly incoherent in character. The optical band gap width of the GdBaCo₂O_5.5 is experimentally determined to be E _g = 0.26 eV, which is close to a theoretical estimate. The type of interband transitions is determined from analyzing the temperature dependence of the optical functions for absorption bands.     

Ellipsometric study of the optical properties of Ca1 − x La x MnO3 single crystals (x = 0–0.2) under n-type doping

The dispersion of the real, ε₁(ω), and imaginary, ε₂(ω), parts of the complex permittivity of Ca_{1 ? x} La_xMnO₃ single crystals (x = 0, 0.05, 0.10, 0.12, 0.20) was studied at room temperature in the spectral region extending from 60 meV to 5 eV. It was found that substitution of lanthanum for calcium shifts the 3.1-eV absorption band in the optical-conductivity spectrum toward higher energies, with the spectral weight of the low-energy wing of the 2.2-eV band becoming redistributed to the band-gap region (E < 1.5 eV) of the starting CaMnO₃ compound. The specific features of optical-conductivity dispersion in the mid-IR region that occur under n-type doping were established. The frequency dependence of optical conductivity was shown to differ from the Drude behavior characteristic of metals. The optical-conductivity spectra of Ca_{1 ? x} La_xMnO₃ were compared with our earlier results on a series of hole-doped La_{1 ? x} Sr_xMnO₃ single crystal.     

Ovoids and Bipartite Subgraphs in Generalized Quadrangles

A point-line incidence system is called an -partial geometry of order (s,t) if each line contains s + 1 points, each point lies on t + 1 lines, and for any point a not lying on a line L, there exist precisely lines passing through a and intersecting L (the notation is pG (s,t)). If = 1, then such a geometry is called a generalized quadrangle and denoted by GQ(s,t). It is established that if a pseudogeometric graph for a generalized quadrangle GQ(s,s ² – s) contains more than two ovoids, then s = 2. It is proved that the point graph of a generalized quadrangle GQ(4,t) contains no K _4,6-subgraphs. Finally, it is shown that if some -subgraph of a pseudogeometric graph for a generalized quadrangle GQ(4,t) contains a triangle, then t 6.     

Optical properties of BiTeI semiconductor with a strong Rashba spin-orbit interaction

We have modeled the 4f ¹-5d ¹ absorption spectrum of a LiYF₄:Ce³⁺ crystal at zero temperature using a microscopic model of the electron-phonon interaction and the real spectrum of LiYF₄ lattice vibrations. Effects caused by mixing of the wave functions of different states of the 5d ¹ excited configuration of the Ce³⁺ ion, which is induced by the electron-phonon interaction, are considered based on the calculations of the second-, third-, and fourth-order exact moments of curvature of the spectrum envelope. We have shown that the large value of the splitting between the maxima of the bands in the absorption spectrum that correspond to transitions to the third and fourth 5d ¹ levels is a result of the nonadiabatic interaction of 5d electrons with lattice vibrations.