Solvable groups having almost relatively prime distinct irreducible character degrees (Q1895556)

Suppose that the finite solvable group \(G\) has the property that for every \(m,n\in\{\chi(1)\mid\chi\in\text{Irr}(G)\}\) with \(m\neq n\), it holds that the greatest common divisor of \(m\) and \(n\) equals either 1 or a prime. Then the main theorem of this paper provides the bound \(\#\{\chi(1)\mid\chi\in\text{Irr}(G)\}\leq 14\). The proof is a little tedious but clear, as it needs a lot of preliminary results all proved here. It is worthwhile to study these results on their own account. There are some theorems derived from the main theorem, for instance when \(K\) is a normal subgroup of \(G\) with \(G/K\) a nonabelian \(p\)-group for some prime \(p\), then the bound 14 can be replaced by 7. The paper deserves a serious study by the interested reader as there is a lot to learn from it.