On finite groups with two irreducible character degrees. (Q1016756)

Suppose \(G\) is a finite group whose character degrees belong to \(\{1,m\}\) where \(m\) is some prescribed integer. Let \(p\geq 3\) be a prime number. Then these \(G\) are classified under the additional condition that \(G\) has precisely \(p\) nonlinear characters. In this review we provide the reader the nilpotent \(G\). Namely, those \(G\) are the groups from a) \(G=C_p\times E\), where \(E\) is extra-special of order \(2^{2c+1}\) (\(c\geq 1\)), and b) \(G\) is a special 2-group of order \(2^{2c}(p+1)\), where \(G/M\cong E\) (as in a)) for each \(M\) a normal subgroup of \(G\) with \(M\) of index 2 in \([G,G]\), and c) \(G\) is dihedral or semihedral or a general quaternion 2-group of order \(4(p+1)\).