Groups with few characters of small degrees (Q1293988)

Let \(\sigma_d(G)\) be the sum of squares of degrees of all those irreducible complex characters whose degrees are not divisible by \(d\); here \(d>1\) divides the order of the finite group \(G\). We have that \(d\) divides \(\sigma_d(G)\). The case where equality occurs, leads to a Frobenius group \(G\) whose kernel has index \(d\) in \(G\). In this paper the case \(\sigma_d(G)=2d\) is studied in some detail; in particular if \(G\) is a 2-group, then it happens to be of maximal class.