On indices of subnormal subgroups of finite soluble groups

All groups considered in this paper are finite soluble groups. Let t(p)(G) = maxHG {j vertical bar p(j) vertical bar vertical bar vertical bar H-G:H vertical bar} and t(G) = max(p is an element of pi(G)) t(p)(G). In this paper, we study the influence of t(G) on the structure of a finite soluble group G. In particular, the hounds of the nilpotent length (derived length, p-length) of a soluble group G with t(G) <= 2 are found.