A well-known theorem of Baer states that if G is a group and \(G/Z_{n}(G)\) is finite, then \( \gamma _{n+1}(G) \) is finite. Kurdachenko et al. proved that if \( G/Z_{n}(G) \) is a locally finite group of finite exponent, then so is \( \gamma _{n+1}(G) \) . In this article, we extend this theorem to groups G with subgroups A of Aut(G) which contain Inn(G) . Furthermore, some new upper bounds of the exponents of \( \gamma _{n+1}(G) \) and \( \gamma _{n+1}(G,A) \) are presented. Moreover we give a proof for the converse of Baer’s theorem considering groups G such that \( G/Z_{n}(G,A) \) and A/Inn(G) are finitely generated or have finite special rank. Finally we conclude that the index of the subgroup \(Z_{n}(G,A)\) is bounded by a precisely determined function in terms of the order of \(\gamma _{n+1}(G,A)\) .
