PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS

Let $\alpha $ be a complex-valued $2$ -cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $ -characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $ -regular are characterised by the values that the irreducible $\alpha $ -characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.