Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

A minimal permutation representation of a group is its faithful permutation representation of least degree. We will find degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, for groups of types E₆, E₇, and E₈. This brings to a close the study of minimal permutation representations of finite simple Chevalley groups. Supported by RFFR grant No. 93-01-01501, through the program “Universities of Russia,” and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 518–530, September–October, 1997.     

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Minimal permutation representations of finite simple exceptional twisted groups

A minimal permutation representation of a group is its faithful permutation representation of least degree. Here the minimal permutation representations of finite simple exceptional twisted groups are studied: their degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, are found. We can thus assert that, modulo the classification of finite simple groups, the aforesaid parameters are known for all finite simple groups. Supported by RFFR grant No. 96-01-01893, through the program “Universities of Russia”, and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 17–35, January–February, 1998.     

Movement of Intransitive Permutation Groups Having Maximum Degree

Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given.     

Two inequalities for parameters of a cellular algebra

Two inequalities are proved. The first is a generalization for cellular algebras of a well- known theorem about the coincidence of the degree and the multiplicity of an irreducible representation of a finite group in its regular representation. The second inequality that is proved for primitive cellular algebras gives an upper bound for the minimal subdegree of a primitive permutation group in terms of the degrees of its irreducible representations in the permutation representation.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 82–95.     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G L₂ (q), where q=p^t for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|³<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L₂(2^t) for some t ≥ 2; moreover, |A|-2^t+1, |B|=2^t, and A is cyclic and B an elementary 2-group. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999.     

Representation of functions of several variables by differences of convex functions

Let Dⁿ be a convex compact set in ℝⁿ. If a function admits a representation of the form f=g−h, where g and are convex and h is bounded from above, then there exists a representation of the same form which is “minimal” in some sense. A recurrent procedure converging to this minimal representation is described. For piecewise-linear functions f (in the cases n=1,2), finite algorithms giving minimal representations are found. A number of examples clarifying some unexpected effects are given. Some problems are formulated. Bibliography: 5 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 36–65. Translated by S. Yu. Pilyugin.     

Lower bounds for the number of conjugacy classes in finite solvable groups

We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|^1/(2^d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G⁽ⁱ⁾: G⁽ⁱ⁺¹⁾] ₁ ^d−1 , where G⁽ⁱ⁾ is theith derived group, leads to a |G|^1/(2d−1) lower bound for k(G), from which we derive a |G|^c/log ₂log₂|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent maximal subgroup, and for all Frobenius groups, solvable or not.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

Large nilpotent subgroups of finite simple groups

Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|²<|G|. Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000.     

Nearly Linear Time Algorithms for Permutation Groups: An Interplay Between Theory and Practice

We survey polynomial time algorithms (both deterministic and random) for computations with permutation groups. Particular emphasis is given to algorithms with running time of the form O(n log ^c |G|), where G is a permutation group of degree n. In the case of small-base groups, i.e., when log |G| is polylogarithmic as a function of n, such algorithms run in nearly linear, O(n log^c' n) time. Important classes of groups, including all permutation representations of simple groups except the alternating ones, as well as most primitive groups, belong to this category. For large n, the majority of practical computations is carried out on small-base groups.In the last section, we present some new nearly linear time algorithms, culminating in the computation of the upper central series in nilpotent groups.     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

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.     

Smooth regularization of plurisubharmonic functions

We consider the problem of approximating a given plurisubharmonic function by smooth plurisubharmonic functions. We propose a new constructive approximation method that permits one to obtain more detailed information about the approximating functions. Thus a functionu ∈ PSH(ℂ ⁿ ) having finite growth order can be approximated by smooth functionsv ∈ PSH(ℂ ⁿ ) so that the difference |v−u| has almost logarithmic growth (Theorem 2). It can also be approximated so that the difference |v−u| has a power-law growth; in this case, however, power-law estimates on |gradv| appear (Theorem 3). Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 312–320, August, 1997. Translated by I. P. Zvyagin     

Attractors for equations describing the flow of generalized Newtonian fluids

Initial boundary-value problems for the equations

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

Concerning the Sierpinski-Schinzel system of Diophantine equations

It is proved that the equation (x ²−1)(y ²−1)=(z ²−1)², |x|≠|y|, |z|≠1, is not solvable in integersx,y,z under the conditionx−y=kz, wherek is a positive integer different from 2. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 181–187, August, 1999.