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

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