Optimal bounds for the Schur index and the realizability of representations

An optimal bound is given for the Schur index of an irreducible complex representation over the field of rational numbers on the class of finite groups of a chosen order or of a chosen exponent. We obtain a sufficient condition for the realizability of an irreducible complex character $\chi$ of a finite group $G$ of exponent $n$ with Schur index $m$, which is either an odd number or has $2$-part no smaller than $4$, over the field of rational numbers in a field $L$ which is a subfield of $\mathbb{Q}(\sqrt[n]{1}\,)$ and $(L:\mathbb{Q}(\chi))=m$. This condition generalizes the well-known Fein condition obtained by him in the case of $n=p^{\alpha}q^{\beta}$. The formulation of the Grunwald-Wang problem on the realizability of representations is generalized, and some sufficient conditions are obtained.
Bibliography: 10 titles.