A converse of Baer's theorem.
Schur showed for any group \(G\) that the finiteness of the quotient group \(G/Z(G)\) implies that of the derived subgroup \(G'\). Baer generalized this to the assertion that the finiteness of \(G/Z_n(G)\) implies that of \(\gamma_{n+1}(G)\). Here, \(\{Z_n\}\) and \(\{\gamma_n\}\) are the upper and lower central series, respectively. Philip Hall proved that the converse of Baer's result is false; he showed that finiteness of \(\gamma_{n+1}(G)\) implies that of \(G/Z_{2n}(G)\) but not necessarily that of \(G/Z_n(G)\). Hekster proved a converse of Baer's theorem under the assumption that \(G\) is finitely generated. In the present paper, the authors prove the stronger result that the converse to Baer's theorem holds under the weaker assumption that \(G/Z_n(G)\) is finitely generated. In fact, they prove the precise version: Theorem. Let \(G\) be a group such that \(G/Z_n(G)\) is finitely generated and \(\gamma_{n+1}(G)\) is finite. Then, \[ |G/Z_n(G)|\leq|\gamma_{n+1}(G)|^{d(G/Z_n(G))}. \] Here, \(d(H)\) denotes the minimal number of generators for a group \(H\). The main idea they employ is that of isoclinism which makes many such proofs simple. For groups \(G\) as in the theorem above which satisfy the additional assumption that \(G\) is nilpotent, the authors deduce the interesting corollary that \(|G/Z_n(G)|\) actually divides \(|\gamma_{n+1}(G)|^{d(G/Z_n(G))}\).
scientific article; zbMATH DE number 6378265