On groups with at most five irrational conjugacy classes

Abstract: Much work has been done to study groups with few rational conjugacy classes or few rational irreducible characters. In this paper we look at the opposite extreme. Let $G$ be a finite group. Given a conjugacy class $K$ of $G$, we say it is irrational if there is some $\chi \in \operatorname{Irr}(G)$ such that $\chi(K) \not \in \mathbb{Q}$. One of our main results shows that, when $G$ contains at most $5$ irrational conjugacy classes, then $|\operatorname{Irr}{\mathbb{Q}}(G)| = |\operatorname{cl}{\mathbb{Q}}(G)|$. This suggests some duality with the known results and open questions on groups with few rational irreducible characters. Our results are independent of the Classification of Finite Simple Groups.