Groups in which all subgroups of infinite rank have bounded near defect

If G is a group with subgroup H and n, m are two fixed non-negative integers, we say that H is (n, m)-subnormal in G if there is a subgroup H_0 containing H such that |H_0:H|≤n and H_0 is subnormal in G with subnormal defect at most m. We define S(n, m) to be the class of all groups of infinite special rank (and the trivial groups) whose subgroups of infinite special rank are (n, m)-subnormal in G. In this paper, we show that S(n, m)-groups are finite-by-nilpotent, in a large class of locally graded groups.