Finite p-groups in which every cyclic subgroup is 2-subnormal

This paper investigates finite p-groups, p greater than or equal to 5, in which every cyclic subgroup has defect at most two. This class of groups is often denoted by U-2,U-1. The main result is a theorem which characterises these groups by identifying a family of groups in U-2,U-1, and showing that any finite p-group in U-2,U-1, with p greater than or equal to 5, must be a homomorphic image of one of these groups.