A characterization of finite nonsimple groups by the set of orders of their elements

It is proved that the permutation wreath product H of a simple Suzuki group Sz(2⁷) and a subgroup fo a symmetric group of degree 23, isomorphic to a Frobenius group of order 253, is (up to isomorphism) distinguished among all finite groups by the set of orders of its elements. Since H possesses a minimal normal subgroup N that contains an element of order equal to the exponent of N, this result furnishes a counterexample to one of the conjectures set forth by Shi [1]. In addition, we show that the direct square of a group Sz(2⁷) is also distinguished by the set of orders of its elements. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 304–322, May–June, 1997.     

Combinatorial optimization problems related to the committee polyhedral separability of finite sets

In the paper, the computational and approximational complexity of the minimal affine separating committee problem, as well as of some important special cases of this problem, is investigated.     

Finite groups of outer automorphisms of free groups

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 5, pp. 82–100, September–October, 1991.     

Recognizability of finite simple groups <Emphasis Type="Italic">L</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) and <Emphasis Type="Italic">U</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) by spectrum

It is proved that finite simple groups L₄(2^m), m ⩾ 2, and U₄(2^m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8. Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry of Education of China (Project for Retaining Foreign Expert). Supported by NSF of Chongqing (CSTC: 2005BB8096). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008.     

Structure of stationary classes of three-value logic functions

This work deals with the properties of transformations of the vectors of values of three-value logic functions to the vectors of the coefficients of their polynomials. A similar transformation of Boolean functions is used in cryptology, and its properties have been thoroughly studied. Stationary classes of three-value logic functions are introduced, and their hierarchy and the exact number of functions in them are obtained. The complete structure of these classes is described.     

Characterization of the finite simple groups by spectrum and order

We give an affirmative answer to Question 12.39 in the Kourovka Notebook. Namely, it is proved that a finite simple group and a finite group having equal orders and same sets of element orders are isomorphic.     

On periodic groups with small orders of elements

We prove that a finite group with the set of element orders equal to {1, 2, 3, 5, 6} is locally finite.