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

The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph

“…On the other hand, many authors have been inspired by looking at connections between characters and conjugacy classes, and, of course, between real characters and real classes (see [2,[22][23][24]). Our works may shed some light on this research.…”

Erratum to: Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups

“…The mistake in the proof of Theorem 1 of [1] comes from the last paragraph of p. 252, in which it was asserted that if C is the conjugacy class of a in G such that C 2 = (C −1 ) 2 , then χ(a)χ(a) is real for each irreducible character χ of G. As a counterexample to this claim, take the above example and the character χ ∈ Irr(G) that takes the following values: χ(1) = 3, χ(a) = α + α 2 + α 4 (α 7 = 1); χ(a 6 ) = α 3 + α 5 + α 6 ; and χ(g) = 0 for every element g ∈ G of order 3. It follows from Problem 3.12 of [2] that…”

“…In [1], the author described real-imaginary conjugacy classes and real-imaginary characters. In Theorem 1 of [1], he showed that C is a real-imaginary conjugacy class of G if and only if C 2 = (C −1 ) 2 . In the next example, we show that this statement is not true.…”