A direct product coming from a particular set of character degrees. (Q2882815)

It is proved that if \(G\) is a solvable group with \(\text{cd}(G)=\{1,m,p,q,mp,mq\}\), where \(p\) and \(q\) are distinct primes and \(m>1\) is an integer not divisible by \(p\) or \(q\), then \(G=A\times B\), where \(\text{cd}(A)=\{1,p,q\}\) and \(\text{cd}(B)=\{1,m\}\). This generalizes [\textit{M. L. Lewis}, J. Algebra 206, No. 1, 235-260 (1998; Zbl 0915.20003), Theorem A]. Here \(\text{cd}(G)\) stands for the set of the degrees of all the irreducible complex characters of \(G\). Lewis in his Theorem A did considers the case \(m\) is a prime, different from \(p\) and \(q\). The proof of the main theorem of the author established here uses a lot of Brauer character theory, Frobenius action on vector spaces, degree graphs and Frobenius groups. As such, the proof of the main theorem is certainly not an easy thing to overcome; it is very nice.