\(p\)-groups with a small number of character degrees and their normal subgroups (Q6645109)

Assume that for any normal subgroup \(N\) of a finite \(p\)-group \(G\) either \(G'\leq N\) or \(N\leq \mathrm{Z}(G)\). The authors show that if \(|G/\mathrm{Z}(G)|=p^{2n+1}\), then \(cd(G)=\{1,p^n\}\). Moreover, they prove that if \(G\) is a finite \(p\)-group with nilpotency class not equal to \(3\), \(|G/\mathrm{Z}(G)|=p^{2n}\), and for any normal subgroup \(N\) of \(G\) either \(G'\leq N\) or \(|N\mathrm{Z}(G)/\mathrm{Z}(G)|\leq p\), then \(cd(G)\subseteq \{1,p^{n-1},p^n\}\).