Some criteria for the decomposability of finite groups

In this paper we prove the following fundamental results.Theorem 1: A finite unsolvable group, every involution of which is contained in a proper isolated subgroup, is decomposable.Theorem 2: Suppose the finite unsolvable group G contains a strongly isolated subgroup M of odd order with isolated normalizer N(M) of even order. If ¦N(M) (M) ¦ > 2, the group G is isomorphic with one of the groups: 1) PSL(2, q), q odd; 2) PGL (2, q), qodd.Translated from Matematicheski Zametki, Vol. 12, No. 6, pp. 717–725, December, 1972.     

Complementability conditions for a periodic almost solvable subgroup in the group containing it

It is proved that if every prime Sylow subgroup of a periodic almost solvable (more generally, periodic W₀-) subgroup H of a group G has a complement in G and if, moreover, H is at most countable and the set (H) is finite, the subgroup H itself possesses a complement in G.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 822–826, June, 1992.     

Intrinsic metrics for lattice ordered groups

Swamy and Jakubik studied the metric ¦x y¦ on lattice ordered groups, and isometries which presere it. We show the only intrinsic metrics on lattice ordered groups are the multiplesn ¦x–y ¦ of theirs, and that the triangle inequality is satisfied by such a metric iff the group is abelian. We show that there are isometries for each of these metrics, but they are rare. We give a simpler proof via permutation groups of the following augmented version of a theorem of Jakubik. IfT is an isometry of the lattice ordered groupG with respect to the metric ¦x¥¦ andT(0)=0, thenG=AB, B is abelian, andT(a+b)=a–b; conversely, any suchT is an isometry.To Paul Conrad on his 60th birthdayPresented by L. Fuchs.     

Groups with a centralizer of sixth order

Let G be a finite fusion-simple group with a self-centralizing subgroup A of sixth order. It is proved that if the centralizer of the involution from A is an unsolvable subgroup of G of an odd index, then G is isomorphic with the Janko group J₁.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 153–159, July, 1977.The author thanks A. I. Starostin for the formulation of the problem and guidance.     

A class of Dirichlet series

Two theorems are proved on Dirichlet series of a special type related to multiplicative functions f(n), ¦f(n)¦1. These theorems develop well-known results of Halász (G. Halász, Acta Math. Acad. Sci. Hungar.,19, Nos. 3–4, 365#x2013;404 (1968)).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 187–194, 1975.     

On the product of finitely generated Abelian groups

It is shown that if a group G is a product of Abelian subgroups A and B one of which is finitely generated, then the group G will have a nontrivial normal subgroup that is contained either in A, or in B.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 443–446, March, 1973.     

On conjectures of Olsson,Brauer, and Alperin

Let G be a finite group, p a prime number, B a p-block of the group G, k(B) the number of irreducible complex characters of R belonging to B, k₀(B) the number of irreducible characters of height zero in B, and let D be the defect group of B. This article considers the relationship between Brauer's conjecture (k(B) ¦D¦), Olsson's conjecture (k₀(B) ¦D/D'¦), and Alperin's conjecture (k₀(B) = k₀(~B), where ~B is a p-block N_G(D) such that ~B^G = B). In particular, Olsson's conjecture is proved for p-blocks for those p-solvable groups G for which a Hall p-subgroup of the group N_G(D) is either supersolvable or has odd order.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 32–35, July, 1992.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

On the smallest limited snake of unit disks

A limited snake of size n is a set of nonoverlapping unit disks D ₁, ..., D _nwith centers c ₁, ..., c _nwhere the distances ¦c _i c _j¦=2 if and only if ¦i–j¦=1, and no disk can touch D ₁ or D _nwithout further common points with D ₁, ..., D _nThe size of the smallest limited snake is proved to be 10.     

Intrinsic metric preserving maps on partially ordered groups

In [11] Pap proved that a surjective mapf from an abelian lattice ordered groupG ₁ onto an abelian Archimedean lattice ordered group G₂ which preserves non-zero intrinsic metricsd ₁, andd ₂ onG ₁ andG ₂, respectively (i.e.d ₁(x,y)=d₁(z, t) implies d₂(f(x)f(y))= d₂(f(z),f(t))) and satisfiesf(0)=0 is a homomorphism and put the question whether that assertion is true in the case that G₂ is a non-Archimedean lattice ordered group. In this paper it is proved that a surjective map from an abelian directedG ₁ onto a directed group G₂ such thatf(0)=0 is a homomorphism if ¦x –y ¦=¦z – t¦ implies ¦f(x) –f(y)¦=¦f(z) –f(t)¦ and it is shown that the answer to the question of Pap is positive.Presented by M. Henriksen.     

Algorithm of shifts for optimal structural schedules

The following problem is solved. Multiset J is given. Find the permutation of its elements that minimizes the value of functional ƒ defined on all permutations of multiset J. Restrictions on functional ƒ are formulated under which the solution is obtained with the aid of the algorithm of shifts with complexity O(¦B(J)¦²¦J¦) of computations of the values of functional ƒ. It is proved that functional satisfies the formulated conditions, and the algorithm of shifts is applicable to it.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 73–92, 1983.     

Infinite p-groups containing exactly p2 solutions of the equation xP=1

We study arbitrary infinite 2-groups with three involutions and infinite locally finite p-groups (p2), containing p²–1 elements of order p. For odd p the groupG=a, where A is a direct product of two quasicyclic 3-groups ¦b¦=9, b³A, and subgroup A is generated by the elements of the commutator ladder of element b, is a unique infinite non-Abelian locally finite p-group whose equation xP=1 has p² solutions.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 11–18, July, 1976.     

A remark on blocking sets of almost Rédei type

In [1] it is shown that ifB is a minimum size blocking set in PG(2,p),p prime, such that there is a lineL meetingB in at least s|B|–p–p+45/20 points, thenB/L consists ofp, p + 1 orp + 2 points. It has been known for a long time that for anyp > 2 there is a unique example with ¦B/L¦= p [2]. In [3] the authors prove that ¦B/L¦ =p + 1 can only occur whenp 7. Here we show that if¦B/L¦= p+2, thenp = 3, 5 or 7, and all examples are classical ones. Besides combinatorial arguments we use polynomials over finite fields and a formula that generalizes the Newton formulae relating power sums and elementary symmetric polynomials.Research of the author was supported by OTKA Grants F-016302, T-019367 and COST Grant 3314/96.     

Conjugate biprimitive finite groups saturated with finite simple subgroups

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L₂(pⁿ)}, {Re(3²ⁿ⁺¹)}, and {Sz(2²ⁿ⁺¹)}. It is proved thai such groups are all isomorphic to {L₂(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 224–245, March–April, 1998.     

Demushkin groups

Let G be a finitely generated pro-p-group with d=d_G>1 generators and a single defining relation. Then, if G is not a Demushkin group, G contains a maximal subgroup with more than p (d–2)+2 generators.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 121–126, July, 1973.     

Locally nilpotent groups with the centralizers satisfying a finiteness condition

Locally nilpotent groups, in which the centralizer of a certain finitely generated subgroup satisfies a certain finiteness condition, are studied. It is proved that if a locally nilpotent group contains a finitely generated subgroup F such that C_G(F) has finite rank, then the center of G is nontrivial.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1084–1087, July–August, 1991.     

On the normalizer condition

It is proved that a torsion-free group , in which every infinite proper subgroup is distinct from its normalizer satisfies the normalizer condition, i.e., every proper subgroup is distinct from its normalizer.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 45–50, January, 1968.     

幂零Hall π-子群的存在性

本文首先将Ｈａｌ定理推广为：设Ｎ为Ｇ的正规子群，若Ｎ为Ｅｎπ群，Ｇ／Ｎ为Ｄπ群，则Ｇ为Ｄπ群．在此基础上得到了群Ｇ为Ｅｎπ群的充要条件为：（１）Ｇ存在正规子群Ｎ，满足Ｎ及Ｇ／Ｎ为Ｅｎπ群；（２）对任意ｐ∈π，任意ｑ∈π ｛ｐ｝及任意ｐ 元素ｘ，ＣＧ（ｘ）含Ｇ的Ｓｙｌｏｗｑ 子群．另外，我们对非Ａｂｌｅ单群的情形也进行了一些讨论．     

Approximative compactness of the algebraic sum of sets

Let X be a group with an invariant metric, A and B nonempty subsets of X with B compact. It is proved that if A is an existence set [1] (approximatively compact [2]) then A + B and B + A are existence sets (approximatively compact). An example is given of a one-dimensional linear metric space (with an invariant metric) in which there exist an approximatively compact set A and an element v such that A + v is not an existence set.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 55–60, January, 1978.