Groups with two extreme character degrees and their normal subgroups (Q2706608)

Let \(G\) be a finite group and \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). The following results are proven.NEWLINENEWLINENEWLINETheorem A. For a non-Abelian \(p\)-group \(G\) the following conditions are equivalent: (i) \(\text{cd}(G)=\{1,|G:Z(G)|^{1/2}\}\). (ii) The set of conjugacy class sizes of \(G\) is \(\{1,|G'|\}\). (iii) \(G'=[x,G]\) for all \(x\in G-Z(G)\). (iv) \(Z(G/N)=Z(G)/N\) for any normal subgroup \(N\) of \(G\) such that \(G'\nleq N\).NEWLINENEWLINENEWLINETheorem B. Let \(G\) be a \(p\)-group such that \(|G:Z(G)|=p^{2n}\) is a square. Then the following statements are equivalent: (i) \(\text{cd}(G)=\{1,p^n\}\). (ii) The normal subgroups of \(G\) either contain \(G'\) or are contained in \(Z(G)\).NEWLINENEWLINENEWLINETheorem C. Let \(G\) be a \(p\)-group such that \(|G:Z(G)|=p^{2n+1}\) is not a square. Then the following statements are equivalent: (i) \(\text{cd}(G)=\{1,p^n\}\). (ii) For any \(N\trianglelefteq G\), either \(G'\leq N\) or \(|NZ(G):Z(G)|\leq p\).NEWLINENEWLINENEWLINESome related results are also presented.