On Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.     

On Some Ideals in the Free Nonassociative Algebra

The free Lie, right, and left Leibniz algebras are obtained as quotients of the free nonassociative algebra by suitable ideals. In this article, we prove some remarkable properties of these ideals.     

On Isomorphisms and Invariants of Finite Dimensional Complex Filiform Leibniz Algebras

In this article, we propose an approach classifying a class of filiform Leibniz algebras. The approach is based on algebraic invariants. The method allows to classify all filiform Leibniz algebras (including filiform Lie algebras) in a given fixed dimensional case.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Correspondences Between 2-Brauer Characters of Solvable Groups

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N _G (P).     

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).