On groups whose factor-group modulo the hypercentre has finite section p-rank

Several theorems of the following type are obtained: Let G be a locally generalized radical group and let p   be a prime. Suppose that G/ζk(G)G/ζk(G) has finite section p-rank at most r  . Then γk+1(G)γk+1(G) has finite section p  -rank. Moreover there exists a function τ(r,k)τ(r,k) such that rp(γk+1(G))≤τ(r,k)rp(γk+1(G))≤τ(r,k).