Subgroups like Wielandt's in soluble groups

For each $m\geq 1$, $u_{m}(G)$ is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most $m$ in $G$.\newline An ascending chain of subgroups $u_{m,i}(G)$ is defined by setting $u_{m,i}(G)/u_{m,i-1}(G)=u_{m}(G/u_{m,i-1}(G))$. If $u_{m,n}(G)=G$ for some integer $n$, the $m$-Wielandt length of $G$ is the minimal of such $n$. In [R. A. Bryce. Subgroups like Wielandt's in finite soluble groups. Math. Proc. Camb. Phil. Soc. 107 (1990), 239-259], Bryce examined the structure of a finite soluble group with given $m$-Wielandt length, in terms of its polynilpotent structure. In this paper we extend his results to infinite soluble groups.