Central extensions of free periodic groups

S. I. Adian ; V. S. Atabekyan

Sbornik. Mathematics, Tome 209 (2018) no. 12, pp. 1677-1689

Résumé

It is proved that any countable abelian group $D$ can be embedded as a centre into a $m$-generated group $A$ such that the quotient group $A/D$ is isomorphic to the free Burnside group $B(m,n)$ of rank $m>1$ and of odd period $n\geqslant665$. The proof is based on a certain modification of the method that was used by Adian in his monograph in 1975 for a positive solution of Kontorovich's famous problem from the Kourovka Notebook on the existence of a finitely generated noncommutative analogue of the additive group of rational numbers with any number $m>1$ of generators (in contrast to the abelian case). More precisely, he proved that the desired analogues in which the intersection of any two non-trivial subgroups is infinite, can be constructed as a central extension of the free Burnside group $B (m, n)$, where $m> 1$, and $n\geqslant665$ is an odd number, using the infinite cyclic group as its centre. The paper also discusses other applications of the proposed generalization of Adian's technique. In particular, the free groups of the variety defined by the identity $[x^n,y]=1$ and the Schur multipliers of the free Burnside groups $B(m,n)$ for any odd $n\geqslant665$ are described. Bibliography: 14 titles.

Détail

Keywords: free Burnside group, central extension, additive group of rational numbers
Mots-clés : Schur multiplier.