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 $G$.