Finite \(p\)-groups with few normal subgroups (Q1590245)

A group \(G\) is called an FNS-group if the normal subgroups of \(G\) either contain \(G'\) or are contained in \(Z(G)\). The author focuses on the finite nonabelian \(p\)-groups having this property; these \(p\)-groups have nilpotency class at most three. The finite \(p\)-groups of class three which are FNS-groups are classified: there are three such groups if \(p=2\), four groups of order \(p^4\) and \(p+7\) groups of order \(p^5\) if \(p\) is odd. The author points out that the class two case is much more complicated because of the multitude of candidates.