Finite p-groups of exponent pe all of whose cyclic subgroups of order pe are normal

Here we classify finite nonabelian p-groups G of exponent pe, e≥2, all of whose cyclic subgroups of order pe are normal in G. Let G0 be the subgroup of G generated by all elements of order pe. If p>2, then G0=G and G is of class 2 (Theorem 1). However if p=2, then |G:G0|≤2 and in case |G:G0|=2 the structure of G is more complicated (Theorems 3, 4 and 5).