TITLE: $p$-groups in which all normal abelian subgroups are cyclic
QUESTION [6 upvotes]: It is well-known that any finite $p$-group in which all its abelian subgroups are cyclic is either a cyclic group or a generalized quaternion group. 
What can be said about $p$-groups in which every normal abelian subgroup is cyclic?

REPLY [6 votes]: See Gorenstein, Finite Groups, Chapter 5, Theorem 4.10. Such groups are as follows:

if $p$ is odd, then $G$ is cyclic;
if $p=2$, then $G$ is either cyclic, or generalised quaternion of order $2^l$ for some $l\geq 3$, or dihedral of order $2^m$ for some $m\geq 4$ (note that $D_8$ does not satisfy the hypothesis), or semidihedral of order $2^n$ for some $n\geq 4$.