Solvable \(D_2\)-groups (Q520414)

A finite group \(G\) is a \(D_{n}\)-group if and only if the number of non-linear irreducible characters of \(G\) is exactly \(n\) less than the number of their different degrees. \textit{Y. Berkovich} et al. [Proc. Am. Math. Soc. 115, No. 4, 955--959 (1992; Zbl 0822.20004)] classified \(D_{0}\)-groups and \textit{Y. Berkovich} and \textit{L. Kazarin} [J. Algebra 184, No. 2, 538--560 (1996; Zbl 0861.20009)] and \textit{Y. Berkovich} [J. Algebra 184, No. 2, 584--603 (1996; Zbl 0861.20008)] classified \(D_{1}\)-groups. The paper under review focuses on the classification of \(D_{2}\)-groups. The authors provide a reference for handling non-solvable \(D_{2}\)-groups (they are: \(S_{5}\), \(A_{6}\), \(A_{7}\), \(A_{9}\), \(M_{22}\) and \(A_{10}\)) and another reference for handling solvable \(D_{2}\)-groups that have two distinct degrees larger than \(1\) with two irreducible characters of that degree for each of them. This leaves them with the case of finite solvable groups such that there exists some \(k>1\) such that \(G\) has exactly \(3\) irreducible characters of degree \(k\), and, for each \(\chi \in \mathrm{Irr}(G)\), if \(\chi(1) \not \in \{1,k\}\), then \(G\) has no other irreducible character of degree \(\chi(1)\). The authors claim to prove that such groups can only exist if they belong to one of twelve listed possibilities.