Some finite p-groups with bounded index of every cyclic subgroup in its normal closure

Let G be a finite p-group. G is called a BI(pm)-group if |G〈a〉:〈a〉|⩽pm for every a∈G. In this paper, we mainly study the properties of BI(2)-groups and BI(p2)-groups for p⩾3, and we study some BI(pm)-groups (m⩾3) with some additional requirement.