Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups

Let G be a finite group. A character chi of G is said to be real-imaginary if its values are real or purely imaginary. A conjugacy class C of a in G is real-imaginary if and only if chi(a) is real or purely imaginary for all irreducible characters chi of G. A finite group G is called real-imaginary if all of its irreducible characters are real-imaginary. In this paper, we describe real-imaginary conjugacy classes and irreducible characters and study some results related to the real-imaginary groups. Moreover, we investigate some connections between the structure of group G and both the set of all the real-imaginary irreducible characters of G and the set of all the real-imaginary conjugacy classes of G.