On the verbal width of finitely generated pro-$p$ groups

On the verbal width of finitely generated pro-p groups

Let p be a prime. It is proved that a non-trivial word w from a free group F has finite width in every finitely generated pro-p group if and only if w 6∈ (F ′)pF ′′. Also it is shown that any word w has finite width in a compact p-adic group.

Finitely Generated Infinite Simple Groups of Infinite Commutator Width

It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can be constructed with decidable word and conjugacy problems.

On the Conjugacy Separability in the Class of Finite P -groups of Finitely Generated Nilpotent Groups

It is proved that for any prime p a finitely generated nilpotent group is conjugacy separable in the class of finite p-groups if and only if the tor-sion subgroup of it is a finite p-group and the quotient group by the torsion subgroup is abelian. 1. Let K be a class of groups. A group G is called residual K (or K-residual) if for each non-unit element a ∈ G there is a homomorphism ϕ of G onto ...

Finitely Generated Abelian Groups

Finitely Generated Semiautomatic Groups

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.