On the theorem of Mann about p-groups all of whose nonnormal subgroups are elementary abelian

We show that if all nonnormal subgroups of a non-Dedekindian p-group G are elementary abelian, then |G′| = p, unless G = D16 is dihedral of order 16. We also describe the p-groups of exponent > p > 2 all of whose nonnormal subgroups have exponent p. We also present a new proof of Passman’s theorem [Ber1, Theorem 1.25] classifying the non-Dedekindian p-groups all of whose nonnormal subgroups have order p. A number of related problems is stated.