On permutizers of subgroups of finite groups

Finite groups with given systems of permuteral and strongly permuteral subgroups are studied. New characterizations of w-supersoluble and supersoluble groups are received.Comment: 11 page

Sub-Riemannian and sub-Lorentzian geometry on \SU(1,1) and on its universal cover

We study sub-Riemannian and sub-Lorentzian geometry on the Lie group \SU(1,1) and on its universal cover \CSU(1,1). In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both \SU(1,1) and \CSU(1,1), connecting two fixed points. In particular, we prove that there is a strong connection between the conjugate loci and the number of geodesics. In the sub-Lorentzian case, we describe the geodesics connecting two points on \CSU(1,1), and compare them with Lorentzian ones. It turns out that the reachable sets for Lorentzian and sub-Lorentzian normal geodesics intersect but are not included one to the other. A description of the timelike future is obtained and compared in the Lorentzian and sub-Lorentzain cases.Comment: 39 pages, 4 figure

Arithmetic graphs of finite groups

In this paper we introduced an arithmetic graph function which associates with every group G the directed graph whose vertices corresponds to the divisors of |G|. With the help of such functions we introduced arithmetic graphs of classes of groups, in particular of hereditary saturated formations. We formulated the problem of the recognition of classes of groups by arithmetic graph functions and investigated this problem for some arithmetic graph functions

On partially conjugate-permutable subgroups of finite groups

Let $R$ be a subset of a group $G$. We call a subgroup $H$ of $G$ the $R$-conjugate-permutable subgroup of $G$, if $HH^{x}=H^{x}H$ for all $x\in R$. This concept is a generalization of conjugate-permutable subgroups introduced by T. Foguel. Our work focuses on the influence of $R$-conjugate-permutable subgroups on the structure of finite groups in case when $R$ is the Fitting subgroup or its generalizations $F^{*}(G)$ (introduced by H. Bender in 1970) and $\tilde{F}(G)$ (introduced by P. Shmid 1972). We obtain a new criteria for nilpotency and supersolubility of finite groups which generalize some well known results

The graph of atomic divisors and constructive recognition of finite simple groups

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of elements of $G$. We present a polynomial-time algorithm that, given a finite set $\mathcal M$ of positive integers, outputs either an empty set or a finite simple group $G$. In the former case, there is no finite simple group $H$ with $\mathcal{M}=\omega(H)$, while in the latter case, $\mathcal{M}\subseteq\omega(G)$ and $\mathcal{M}\neq\omega(H)$ for all finite simple groups $H$ with $\omega(H)\neq\omega(G)$

On the structure of finite groups isospectral to finite simple groups

Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group $L$ is said to be almost recognizable by spectrum if every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$. It is known that all finite simple sporadic, alternating and exceptional groups of Lie type, except $J_2$, $A_6$, $A_{10}$ and $^3D_4(2)$, are almost recognizable by spectrum. The present paper is the final step in the proof of the following conjecture due to V.D. Mazurov: there exists a positive integer $d_0$ such that every finite simple classical group of dimension larger than $d_0$ is almost recognizable by spectrum. Namely, we prove that a nonabelian composition factor of a~finite group isospectral to a finite simple symplectic or orthogonal group $L$ of dimension at least 10, is either isomorphic to $L$ or not a group of Lie type in the same characteristic as $L$, and combining this result with earlier work, we deduce that Mazurov's conjecture holds with $d_0=60$.Comment: 13 page

Generalized Fitting subgroups of finite groups

In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$ and its generalizations: the quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq \Phi(G)$ and $\tilde{F}(G)/\Phi(G)=Soc(G/\Phi(G))$. We sum up known properties of $\tilde{F}(G)$ and suggest some new ones. Let $R$ be a subgroup of a group $G$. We shall call a subgroup $H$ of $G$ the $R$-subnormal subgroup if $H$ is subnormal in $\langle H,R\rangle$. In this work the influence of $R$-subnormal subgroups (maximal, Sylow, cyclic primary) on the structure of finite groups are studied in the case when $R\in\{F(G), F^*(G),\tilde{F}(G)\}$

Boundary distortion estimates for holomorphic maps

We establish some estimates of the the angular derivatives from below for holomorphic self-maps of the unit disk at one and two fixed points of the unit circle provided there is no fixed point inside the unit disk. The results complement Cowen-Pommerenke and Anderson-Vasil'ev type estimates in the case of univalent functions. We use the method of extremal length and propose a new semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to receive a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary

Renormalization group, operator product expansion and anomalous scaling in models of passive turbulent advection

The field theoretic renormalization group is applied to Kraichnan's model of a passive scalar quantity advected by the Gaussian velocity field with the pair correlation function $\propto\delta(t-t')/k^{d+\epsilon}$. Inertial-range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of ``dangerous'' composite operators (powers of the local dissipation rate), whose {\it negative} critical dimensions determine anomalous exponents. The latter are calculated to order $\epsilon^3$ of the $\epsilon$ expansion (three-loop approximation).Comment: 4 page

Effect of neutron irradiation on the properties of FeSe compound in superconducting and normal states

Effect of atomic disordering induced by irradiation with fast neutrons on the properties of the normal and superconducting states of polycrystalline samples FeSe has been studied. The irradiation with fast neutrons of fluencies up to 1.25\cdot10^20 cm^-2 at the irradiation temperature Tirr ~ 50 \degree C results in relatively small changes in the temperature of the superconducting transition T_c and electrical resistivity Rho_25. Such a behavior is considered to be traceable to rather low, with respect to that possible at a given irradiation temperature, concentration of radiation defects, which is caused by a simpler crystal structure, considered to other layered compounds.Comment: 3 pages, 3 figure