\(p\)-groups with exactly four codegrees (Q2208332)

Let \(G\) be a finite group. For \(\chi\in\mathrm{Irr}(G\)) the codegree is defined as \(\mathrm{cod}(\chi)=|G:\mathrm{Ker}(\chi)|/\chi(1)\). Write \(\mathrm{cod}(G)\) for the set of all codegrees of \(G\). It is known that a \(p\)-group with exactly two codegrees is elementary abelian, and a \(p\)-group with exactly three codegrees has nilpotence class at most 2. The paper under review makes some advances on bounding the nilpotence class of \(p\)-groups with four codegrees by proving the following result. Theorem. Suppose that \(\mathrm{cod}(G)=\{1, p, p^b, p^a\}\), where \({2\le b< a}\). If any of the following holds, then \(G\) has nilpotence class at most 4: (i) \(|\mathrm{cd}(G)|=2\); (ii) \(\mathrm{cd}(G)=\{1,p,p^2\}\); (iii) \(|G:G^{'}|=p^2\). There are some more results under more special hypotheses.