A well-known theorem of Baer states that in a given group G, the \( (n+1) \) th term of the lower central series of G is finite when the index of the nth term of the upper central series is finite. Recently, Kurdachenko and Otal proved a similar statement for this theorem when the upper hypercenter factor of a locally generalized radical group has finite special rank. In this paper, we first decrease the Ellis’ bound obtained for the order of \(\gamma _{n+1}(G).\) Then we extend Kurdachenko’s result for locally generalized radical groups. Moreover, some new upper bounds for the special rank of \( \gamma _{n+1}(G,A) \) are also given, where A is a subgroup of automorphisms of G which contains inner automorphisms of G.
