O. Macedońska

Automorphisms of relatively free nilpotent groups of infinite rank

Let F be a free group and V a characteristic subgroup of F. Then the natural homomorphism from F to F/V gives rise to a homomorphism 1: Aut(F) + Aut(F/I’) from the automorphism group of F to the automorphism group of F/V. In this paper we shall prove that x is surjective provided that F has infinite rank and F/V is nilpotent.

On Positive Law Problems in the Class of Locally Graded Groups

Abstract We consider five known problems concerning positive laws in groups. One of them has a counterexample, the others are open in general. It is shown that three of the problems are equivalent and all of them have a positive solution in the class of locally graded groups.

All automorphisms of the 3-nilpotent free group of countably infinite rank can be lifted

Abstract It is known from S. Andreadakis (Proc. London Math. Soc. (3) 15 (1965), 239–268) and S. Bachmuth (Trans. Amer. Math. Soc. 122 (1966), 1–17) that for a free 3-nilpotent group F of rank n the map Aut F → Aut F is not onto. It is proved here that for F countably infinitely generated the map is onto.

A Note on Groups of Intermediate Growth

It was proved by Grigorchuk (1983) that there exist groups which are neither of polynomial nor of exponential growth. Their growth is called “intermediate”. We show that every group of intermediate growth has either a residually finite quotient of intermediate growth or a simple section of intermediate growth.

Collapsing groups and positive laws

The paper concerns the question of A. Shalev: is it true that every collapsing group satisfies a positive law? We give a positive answer for groups in a large class C, including all soluble and residually finite groups.

A defining property of virtually nilpotent groups

We answer the question: which property distinguishes the virtually nilpotent groups among the locally graded groups? The common property of each finitely generated group to have a finitely generated commutator subgroup is not sufficient. However, the finitely generated commutator subgroup of F2(varG), a free group of rank 2 in the variety defined by G, is the necessary and sufficient condition.

ON VARIETIES OF GROUPS WITHOUT POSITIVE LAWS

ABSTRACT We give negative answers to three questions concerning positive laws and varieties.

GB-Problem in the Class of Locally Graded Groups

A problem, we consider, is equivalent to the one posed in 1981 by Bergman: Let G be a group and S a subsemigroup of G which generates G as a group. Must each identity satisfied in S be satisfied in G? The first counterexample was found in 2005 by Ivanov and Storozhev. It gives a negative answer to the problem in general. However we show that the problem has an affirmative answer for locally residually finite groups and for locally graded groups containing no free noncyclic subsemigroups.

TWO QUESTIONS ON SEMIGROUP LAWS

2. Let a semigroup — la bw impl a y a semigroup law u = v in groups. Doesthe same implication hold in semigroups?To show implication of laws in semigroups we can use only so-called positive endo-morphisms, which map generators to positive words. It is shown in [8] (an example atthe end of this paper), that all implications for positive laws of length ^ 5 which holdin groups, also are valid for semigroups. The fac

On non-Hopfian groups of fractions

Abstract The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.