Finite groups with some bounded codegrees (Q6596315)

Let \(\chi\) be an irreducible character of a finite group \(G\). The codegree \(\operatorname{cod}(\chi)\) of \(\chi\) is defined as the number \(|G : \ker \chi| / \chi(1)\), where \(\ker \chi\) is the kernel of \(\chi\). As has been shown in numerous research papers, the set of codegrees of a finite group significantly impacts its structure. Imposing certain conditions on the set of codegrees can determine some of the properties of the group in general. In the paper under review, the authors investigate a solvability criterion based on the minimum of \(\chi(1)^2 / \operatorname{cod}(\chi)\), where \(\chi\) ranges over all irreducible characters of \(G\). In particular, they prove that if\N\[\N\frac{\chi(1)^2}{\operatorname{cod}(\chi)} > \frac{2^9 \cdot 3^2 \cdot 19^2}{5 \cdot 7^3 \cdot 11 \cdot 13}\N\]\Nfor every irreducible character \(\chi\) of \(G\), then \(G\) is solvable. The proof relies on the classification of finite simple groups.