Correction to: Monatshefte für Mathematik (2023) 201:349–358 https://doi.org/10.1007/s00605-022-01726-w

Theoren A(b) of [1] asserts that if G is a finite group, N is a normal subgroup of G and K is a conjugacy class of G such that \(K \cup K^{-1}\) is a union of cosets of N, then K is a real-imaginary class. This means, following [2], that every irreducible character of G takes real or purely-imaginary values at K. This assertion, however, is only true whenever we assume further that K is not a union of cosets of N, and we will give an example to illustrate this. Therefore, the aim of this note is to give a corrected version and a proof of this result, which should be as follows.

FormalPara Theorem A

Suppose that G is a finite group, N is a normal subgroup of G and K is a conjugacy class of G such that \(K \cup K^{-1}\) is a union of cosets of N.

  1. (b)

    If K is not a union of cosets of N, then K is a real-imaginary class, that is, \(\chi (x)\) is real or purely imaginary for every \(\chi \in \textrm{Irr}(G)\).

We remark, however, that this mistake does not affect the rest of the paper. Theorem A(a) and (c) remain true, indeed the proof of (b) appears after (a) and (c) are proved. Likewise, the rest of the derived results in the paper hold since Theorem A(b) is not employed in their proofs. We state and prove Theorem A(b) in a slightly different manner, just as it appears in [1, Theorem 3.3].

FormalPara Theorem

Suppose that G is a finite group, N is a normal subgroup of G and \(K = x^G\) is a conjugacy class of G such that \(xN\subseteq K \cup K^{-1}\).

  1. (b)

    If \(xN\not \subseteq K\), then K is a real-imaginary class, that is, \(\chi (x)\) is real or purely imaginary for every \(\chi \in \textrm{Irr}(G)\).

FormalPara Proof

Mimic the first paragraph of the proof of [1, Theorem 3.3(b)] and assume also that K is non-real, otherwise the result is trivial. Now the hypotheses imply that \(KN\subseteq K\cup K^{-1}\) and also that \(KN\not \subseteq K\). Hence there must exist a conjugacy class C of G contained in N such that \(KC\subseteq K\cup K^{-1}\) but \(KC\not \subseteq K\). Then we can write

$$\begin{aligned} {\widehat{K}}{\widehat{C}}=a{\widehat{K}}+b\widehat{K^{-1}} \quad \textrm{and} \quad {\widehat{K}}\widehat{C^{-1}}=c{\widehat{K}}+d\widehat{K^{-1}} \end{aligned}$$

for some integers a, b, c, d, with \(b, d>0\). Once we have \(b,d\ne 0\), then the rest of the proof of [1, Theorem 3.3(b)] applies, including the assertion \({\widehat{K}}^2 =\widehat{K^{-1}}^2\) in the last paragraph of that proof. Note that if we do not ensure that d and b are distinct from 0 for some conjugacy class C, then we cannot guarantee that K is a real-imaginary class. \(\square \)

To finish we give a family of examples showing that the new hypothesis in Theorem A(b) is really necessary. Suppose that G is a Frobenius group with odd order cyclic complement and kernel N. It is well known and elementary to see that for every \(x\in G{\setminus } N\) the coset xN coincides with the conjugacy class \(x^G\) of x. Thus every conjugacy class \(K=x^G\subseteq G{\setminus } N\) satisfies \(K\cup K^{-1}=xN \cup x^{-1}N\), so it is a union of cosets of N. However, no such class is real-imaginary. In fact, every non-trivial character of G containing N in its kernel takes an odd order root of unity value at K (distinct from 1), which of course neither is real nor purely-imaginary.