Ordinary and p-modular character degrees of solvable groups
Let G be a finite solvable group, let p be a prime and let \(cd(G)=\{\chi (1)|\chi \in Irr_{{\mathbb{C}}}(G)\}\) resp. \(cd_ p(G)=\{\beta (1)|\beta \in IBr_ p(G)\}\) denote the set of irreducible complex resp. p-modular character degrees. Note that \(cd_ p(G)\) is a subset of cd(G) by the Fong-Swan theorem. The paper under review is concerned with the following question: Given arithmetical conditions on \(cd_ p(G)\). What can we say about cd(G). Most of the paper is spent for the proof of the following main result: If \(O_ p(G)=1\) and \(cd_ p(G)\) is squarefree (i.e. \(q^ 2\nmid \beta (1)\) for all primes q and all \(\beta \in IBr_ p(G))\) then cd(G) is cubefree. In this case the group structure is described to some extend very precise. Easy examples show that the bound is best possible. In a final section the author assumes more generally that \(q^ n\nmid \beta (1)\) for all \(\beta (1)\in cd_ p(G)\), all primes q and some \(n\in {\mathbb{N}}\). Functions in n are then given which bound the derived length of G, the Fitting length of G and the largest exponent m such that \(q^ m| \chi (1)\) for some \(\chi \in Irr_{{\mathbb{C}}}(G)\) and some prime q.
scientific article; zbMATH DE number 4175226
scientific article