On circle groups of nilpotent rings of charateristic 2

We prove that every nilpotent group of class 2 and exponent 4 is the circle group of a nilpotent ring of index 3 and characteristic 2.Dedicated to Professor László Fuchs on his 70th birthdayResearch supported by the Hungarian National Foundation for Scientific Research grant No. T 014279     

Structure of a group with elements of order at most 4

We prove that every group in which the order of each element is at most 4 either possesses a nontrivial class 2 nilpotent normal Sylow subgroup or includes a normal elementary abelian 2-subgroup the quotient by which is isomorphic to the nonabelian group of order 6.     

The exponents of nonabelian tensor products of groups

We prove that the exponent of the nonabelian tensor product of two locally finite groups can be bounded in terms of exponents of given groups. Several estimates for the exponents of nonabelian tensor squares are obtained. In particular, if the group G is nilpotent of class ≤3 and of finite exponent, then the exponent of its nonabelian tensor square divides the exponent of G.     

Powerful 2-Engel groups II

We conclude our classification of powerful 2-Engel groups of class three that are minimal in the sense that every proper powerful section is nilpotent of class at most two. In the predecessor to this paper we obtained three families of minimal groups. Here we get a fourth family of minimal examples that is described in terms of irreducible polynomials over the field of three elements. We also get one isolated minimal example of rank 5 and exponent 27. The last one has a related algebraic structure that we call a “symplectic alternating algebra.” To each symplectic alternating algebra over the field of three elements there corresponds a unique 2-Engel group of exponent 27.     

Semivarieties of nilpotent groups

Semivarieties of groups are quasivarieties defined by quasi-identities of the form t = 1 → f = 1. It is proved that a set of semivarieties in every variety of class two nilpotent p-groups of finite exponent having a commutator subgroup of exponent p (p is a prime) is at most countable. It is stated that a variety of class two nilpotent groups with commutator subgroup of exponent p contains a set of semivarieties of the cardinality of the continuum.     

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2). Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996.     

有限秩的可解群的正则自同构

设G是有限秩的剩余有限可解群或是有限秩的剩余有限可解群的有限扩张,α是G的一个索数p阶正则自同构且φ:G→G(g→[g,α])是满射,则G是幂零类不超过h(p)的幂零群,其中h(p)是只与p有关的函数.     

Maximal and Sylow Subgroups of Solvable Finite Groups

The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.     

Nilpotency class of a free group of the product of certain varieties

In this paper we calculate the class of a free group of finite rank of the product of two varieties, where the left factor is the variety of all Abelian groups of exponent pⁿ and the right is the variety of all nilpotent groups of class at most c and exponent p^t (where c < p).Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 91–98, January, 1976.The author is grateful to A. L. Shmel'kin for posing this problem and for his guidance.     

Torsionsgruppen,deren Untergruppen alle subnormal sind

We prove that a group, which is the extension of a nilpotent torsion group by a soluble group of finite exponent and all of whose subgroups are subnormal, is nilpotent. The problem can be easily reduced to the investigation of extensions of abelian torsion groups by elementary abelian p-groups with all subgroups of these extensions subnormal.     

The arithmetical hierarchy of nilpotent torsion-free groups

Previously, N. Khisamiev proved that all {ie172-1} Abelian torsion-free groups are {ie172-2}. We prove that for the class of nilpotent torsion-free groups, the situation is different: even the quotient group F of a {ie172-3} nilpotent group of class 2 by its periodic part may fail to have a {ie172-4}. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 308–313, May–June, 1996.     

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.     

Rational sets in finitely generated nilpotent groups

We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2 whose set of solutions is not rational. Supported by RFFR grant No. 98-01-00932. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 379–394, July–August, 2000.     

The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ?. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.     

Locally nilpotent groups satisfying the weak minimum or maximum condition for subgroups of bounded nilpotency class

We study locally nilpotent groups containing subgroups of classc, c>1, and satisfying the weak maximum condition or the weak minimum condition on c-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group G is either nilpotent or periodic locally nilpotent and if all of its c-nilpotent subgroups are of finite rank, then G is of finite rank. If G is a non-periodic locally nilpotent group, these results, in general, are not valid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 384–389, March, 1992.     

Lie rings admitting an automorphism of order 4 with few fixed points. II

We consider a Lie ring (algebra) L that admits an automorphism φ of order 4 with a finite number m of fixed points (with a fixed-point subalgebra of finite dimension m). It is proved that L contains a subring S of m-bounded index in the additive group L (a subalgebra S of m-bounded codimension), which possesses a nilpotent ideal I of class bounded by some constant, such that the factor-ring S/I is nilpotent of class ≤2. As a consequence, it is proved that, under the same conditions, L has a subring G of m-bounded index in the additive group of L (a subalgebra G of m-bounded codimension), in which an ideal generated by the Lie subring [G, ?²]=«ng?g+g^{? 2} | g∈G»ng (the subalgebra [G, ?²]=«ng?g+g^{? 2} | g∈G»ng is an ideal in G which) is nilpotent of class bounded by some constant (and its factor-algebra G/[G, ?²] is nilpotent of class ≤2 with a derived algebra (square) of m-bounded dimension). In proofs, we use the results of [1] and develop further the version of the method of generalized centralizers employed therein.     

Structure of the automorphism group for partially commutative class two nilpotent groups

Let R be a ring, which is either a ring of integers or a field of zero characteristic. For every finite graph Γ, we construct an R-arithmetic linear group H(Γ). The group H(Γ) is realized as the factor automorphism group of a partially commutative class two nilpotent R-group G _Γ. Also we describe the structure of the entire automorphism group of a partially commutative nilpotent R-group of class two.     

The Generalised Pro-Fitting Subgroup of a Profinite Group

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.     

Powerfully nilpotent groups of maximal powerful class

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.