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

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 $\omega(H)\neq\omega(G)$

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 $R\in\{F(G), F^*(G),\tilde{F}(G)\}$

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

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

Renormalization group in the statistical theory of turbulence: Two-loop approximation

The field theoretic renormalization group is applied to the stochastic Navier--Stokes equation that describes fully developed fluid turbulence. The complete two-loop calculation of the renormalization constant, the beta function and the fixed point is performed. The ultraviolet correction exponent, the Kolmogorov constant and the inertial-range skewness factor are derived to second order of the $\epsilon$ expansion.Comment: 5 page

Shape memory ferromagnets

In ferromagnetic alloys with shape memory large reversible strains can be obtained by rearranging the martensitic domain structure by a magnetic field. Magnetization through displacement of domain walls is possible in the presence of high magnetocrystalline anisotropy, when martensitic structure rearrangement is energetically favorable compared to the reorientation of magnetic moments. In ferromagnetic Heusler alloys Ni$_{2+x}$Mn$_{1-x}$Ga the Curie temperature exceeds the martensitic transformation temperature. The fact that these two temperatures are close to room temperature offers the possibility of magnetically controlling the shape and size of ferromagnets in the martensitic state. In Ni$_{2+x}$Mn$_{1-x}$Ga single crystals, a reversible strain of $\sim 6$% is obtained in fields of $\sim 1$ T.Comment: review on ferromagnetic shape memory alloys (FSMAs

On Order-Disorder (L21→B2′L2_1 \to B2^\prime) Phase Transition in Ni2+xMn1−x_{2+x}Mn_{1-x}Ga Heusler Alloys

Order-disorder phase transition in Ni$_{2+x}Mn_{1-x}$Ga Heusler alloys has been studied. It was found that the $L2_1 \to B2^{\prime}$ phase transition in Ni2+xMn1-xGa (x = 0.16 - 0.20) Heusler alloys is of second order and the temperature of this transition decreases with Ni excess.Comment: 3 pages, 3 figures, revtex