On finite soluble groups with restrictionson cofactors of subgroups that lie in the Fitting subgroup

Properties of finite soluble groups that depend on the orders of cofactors of subgroups that lie in the Fitting subgroup are investigated. Let n be the maximal integer such that for some prime $p$, $p^n$ divides the order of such cofactor. It is proved that the rank and the derived length of the factor group by the Frattini subgroup and the nilpotent length of a finite soluble group are bounded by certain functions of $n$. We construct examples that show that the estimates obtained are exact for groups in which cubes of primes do not divide the orders of cofactors of subgroups that lie in the Fitting subgroup.