AUTOMORPHISMS OF FREE BURNSIDE GROUPS OF PERIOD 3

We have proved that any automorphism of the free Burnside group $ B(3) $ of period 3 and an arbitrary rank is induced by an automorphism of the free group of the same rank.

Characterization of finitely generated groups by types

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups [Formula: see text] are fully characterized in the class of all groups by the set [Formula: see text] of types realized in [Formula: see text]. Furthermore, it turns out that these groups [Formula: see text] are fully characterized already by some particular rather restricted fragments of the types from [Formula: see text]. In particular, every finitely generated nilpotent group is completely defined by its [Formula: see text]-types, while a finitely generated rigid group is completely defined by its [Formula: see text]-types, and a finitely generated metabelian or polycyclic group is completely defined by its [Formula: see text]-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

The automorphisms of endomorphism semigroups of relatively free groups

The question of describing the automorphisms of [Formula: see text] for a free algebra [Formula: see text] in a certain variety was considered by different authors since 2002. In this paper, we consider this question for the class of all relatively free groups having only cyclic centralizers of non-trivial elements. We prove that each automorphism of the endomorphism semigroup [Formula: see text] of groups [Formula: see text] from this class is uniquely determined by its action on the subgroup of inner automorphisms [Formula: see text]. The obtained general result includes the following cases: absolutely free groups, free Burnside groups of odd period [Formula: see text], free groups of infinitely based varieties of Adian (the cardinality of the set of such varieties is continuum), and so on.

A Connected 3-State Reversible Mealy Automaton Cannot Generate an Infinite Burnside Group

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, all such examples have been constructed as groups generated by non-reversible automata. Moreover, it was recently shown that 2-state reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state reversible Mealy automata, using new original techniques. The results rely on a fine analysis of associated orbit trees and a new characterization of the existence of elements of infinite order.