A remark on the converse of Baer’s theorem for Lie algebras

Abstract

The classic theorem of Schur states that for a group $G$ the finiteness of $G/Z(G)$ implies the finiteness of $G'$. Baer extends this theorem and shows that if $G/Z_n(G)$ is finite for a group $G$, the so is $\gamma _{n+1}(G)$, where $Z_n(G)$ and $\gamma _{n+1}(G)$ are the $n$th term of the upper central series and the $(n+1)$th term of lower central series of $G$, respectively. These results are extended to Lie algebras by Salemkar et. al. by showing that the converse of Baer’s theorem remains hold when the corresponding Lie algebra is finitely generated. Generalizing this result, we also obtain a bound for the $n$-central factor of a Lie algebra.