Factorizations of characters and a question of Feit
Let G be a finite group, \(\chi\) a complex character of G, \({\mathbb{Q}}\) the rational field, \({\mathbb{Q}}(\chi)={\mathbb{Q}}(\chi (x)|\) \(x\in G)\), and \({\mathbb{Q}}_ m\) the field of mth roots of unity over \({\mathbb{Q}}\). Let f(X) be the smallest positive integer m wth \({\mathbb{Q}}(\chi)\subseteq {\mathbb{Q}}_ m\). The authors give some affirmative answers to the following question of Feit: If \(\chi\) is an irreducible character of G, is it true that G contains an element of order f(\(\chi)\)? The authors use character factorization to arrive to a number of related results including: Theorem A. Let G be a finite group all of whose non- Abelian simple composition factors are minimal simple. Let \(\chi\) be any irreducible character of G. Then G contains an element of order f(\(\chi)\). Theorem B. Let G be a finite group all of whose composition factors are N-groups. Suppose that \(\chi\) is irreducible and m is some positive integer such that \(m| f(\chi)\). Assume at least one of the following holds: (i) 36\(\nmid \chi (1)\); (ii) \(A_ 7\) is not involved in G (as a composition factor if \(\chi\) is quasi-primitive); (iii) 30\(\nmid m\); (iv) \(G'=G\) and \(\chi\) is quasi-primitive. Then G contains an element of order m.
scientific article; zbMATH DE number 4021424
scientific article