Bounds for the Generalization of Baer's Type Theorems

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.