Solvable groups satisfying the two-prime hypothesis. I.
If \(G\) is a finite group, let \(\text{cd}(G)\) be the set of its irreducible complex character degrees. \(G\) is said to satisfy the \(n\)-prime hypothesis if any two distinct elements from \(\text{cd}(G)\) have at most \(n\) primes in common (counting multiplicities). It is conjectured that there is a function \(f(n)\) such that \(|\text{cd}(G)|\leq f(n)\) for all solvable \(G\) satisfying the \(n\)-prime hypothesis. It is well-known that one can choose \(f(0)=3\), and \(f(1)=9\) was established by \textit{M. L. Lewis} [Algebr. Represent. Theory 8, No. 4, 479-497 (2005; Zbl 1118.20010)]. The paper under review is the first of two papers whose goal is to prove that one can choose \(f(2)=462515\). In this first paper the following result is obtained: If the solvable group \(G\) satisfies the 2-prime hypothesis and has a nonabelian nilpotent factor group, then \(|\text{cd}(G)|\leq 88\). The proof naturally is quite technical. The paper also contains some results on the general \(n\)-prime hypothesis and related problems involving the \(\pi\)-parts of character degrees for a set \(\pi\) of primes.
scientific article; zbMATH DE number 5156930
scientific article