New variations on the theme of Baer’s theorem

Abstract

Abstract Let $$\gamma _s(G)$$ γ s ( G ) and $$Z_s(G)$$ Z s ( G ) denote the s -th terms of the lower and upper central series of a group G , respectively. A classical theorem by R. Baer states that if $$Z_s(G)$$ Z s ( G ) has finite index in G , then $$\gamma _{s+1}(G)$$ γ s + 1 ( G ) is also finite. In this paper, we prove that if G is a generalized soluble group such that $$\gamma _s(G)/(\gamma _s(G) \cap Z_t(G))$$ γ s ( G ) / ( γ s ( G ) ∩ Z t ( G ) ) has finite rank r for some s , t , then the rank of $$\gamma _{s+t}(G)$$ γ s + t ( G ) is finite and ( r , s , t )-bounded. Moreover, a corresponding result replacing the finite-rank assumption by the condition that $$\gamma _s(G)/(\gamma _s(G) \cap Z_t(G))$$ γ s ( G ) / ( γ s ( G ) ∩ Z t ( G ) ) is a Chernikov group of bounded size is also obtained. These results extend recent generalizations of the classical Baer’s theorem.