Waldemar Hołubowski

Parabolic subgroups of Vershik-Kerov’s group

In this note we show that all parabolic subgroups of Vershik-Kerovs group GLB(R) (i.e. subgroups containing T(∞,R)-the group of infinite dimensional upper triangular matrices) are net subgroups for a wide class of semilocal rings R.

Symmetric words in metabelian groups

An n-ary word w(x1,…,xn) is called n-symmetric for a group G if w(g1,…,gn) = w(gσ 1,…,gσ n) for all g1,…,gn in G and all permu¬tations a in the symmetric group Sn. In this note we describe 2 and 3-symmetric words in free metabelian groups and metabelian groups of nilpotency class c, for arbitrary c.

Automorphisms of a free group of infinite rank

The problem of classifying the automorphisms of a free group of infinite countable rank is investigated. Quite a reasonable generating set for the group Aut F∞ is described. Some new subgroups of this group and structural results for them are presented. The main result says that the group of all automorphisms is generated (modulo the IA-automorphisms) by strings and lower triangular automorphisms.

A new measure of growth for groups and algebras

The notion of a bandwidth growth is introduced, which generalizes the growth of groups and the bandwidth dimension, first discussed by J. Hannah and K. C. O’Meara for countable-dimensional algebras. The new measure of growth is based on certain infinite matrix representations and on the notion of growth of nondecreasing functions on the set of natural numbers. Two natural operations are defined on the set Ω of growths. With respect to these operations, Ω forms a lattice with many interesting algebraic properties; for example, Ω is distributive and dense and has uncountable antichains. This new notion of growth is applied in order to define bandwidth growth for subgroups and subalgebras of infinite matrices and to study its properties.

Bijections preserving commutators and automorphisms of unitriangular groups

We complete the characterization of bijection preserving commutators (PC-maps) on the group of unitriangular matrices over a field F, where . PC-maps were recently described up to almost identity PC-maps by Chen et al.[18] for finite n and by Słowik[17] for . An almost identity map is a map fixing elementary transvections. We show that an almost identity PC-map is a multiplication by a central element. In particular, if , then an almost identity map is identity. In view of the result of Słowik, this shows that any PC-map of is an automorphism.

Note on Sus̆kevic̆’s problem on zero divisors

ABSTRACT We solve an old Sus̆kevic̆’s problem on right zero divisors in the ring T(∞,K) of infinite ℕ×ℕ upper triangular matrices over a field K. In the solution we use a notion of strong linear independence.

Derivations of the Lie algebra of infinite strictly upper triangular matrices over a commutative ring

ABSTRACT Let 𝒩(∞,R) be the Lie algebra of infinite strictly upper triangular matrices over a commutative ring R. We show that every derivation of 𝒩(∞,R) is a sum of diagonal and inner derivations.

A note on commutators in the group of infinite triangular matrices over a ring

We investigate the commutators of elements of the group of infinite unitriangular matrices over an associative ring with and a commutative group of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence, we give a complete characterization of the lower central series of the group including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring , we show that the derived subgroup of coincides with the group . These results generalize the results obtained for triangular groups over a field.