Degrees of rational characters of finite groups.
A classical theorem of John Thompson on character degrees states that if the degree of any complex irreducible character of a finite group is 1 or divisible by a prime \(p\), then this group has a normal \(p\)-complement. In this paper, this important result is generalized and extended. We mention here only some results out of the 32 Theorems, Lemmas and Propositions of the paper. The authors state as their main result, the following Theorem A: Let \(G\) be a finite group. Then \(\chi(1)\) is even for each non-linear irreducible \(\mathbb{Q}\)-valued character \(\chi\) of \(G\) if and only if \(G\) has a normal 2-complement. Corollary B: Let \(K\) be the intersection of the kernels of the irreducible \(\mathbb{Q}\)-valued characters of the finite group \(T\). Then \(K\) has a normal 2-complement. Theorem C: Let \(G\) be a finite group, and consider the field \(\mathbb{Q}(\zeta)\), where \(\zeta^p=1\), \(\zeta\neq 1\), \(p\) a given prime number. Then \(G\) has a normal \(p\)-complement if each non-linear \(\mathbb{Q}(\zeta)\)-valued irreducible character \(\chi\) has its degree \(\chi(1)\) divisible by \(p\). The proof of Theorem A is certainly not trivial and it uses the classification of the finite simple groups. The deep part in the proofs of Theorems A and C comes from Theorem 2.3. Let \(G\) be a finite group, \(p\) a prime number. Then \(G\) is solvable if each irreducible \(\mathbb{Q}(\zeta)\)-valued character of \(G\) is linear or has a degree divisible by \(p\). The whole paper contains many interesting new and intricate results. Fantastic stuff, anyway.
scientific article; zbMATH DE number 5712556
scientific article