On groups with at most five irrational conjugacy classes

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 $\chi \in \mathrm{Irr}(G)$ such that $\chi (K)\in ̸\mathbb{Q}$. One of our main results shows that, when $G$ contains at most $5$ irrational conjugacy classes, then $\mathrm{\mid }{\mathrm{Irr}}_{\mathbb{Q}}(G)\mathrm{\mid }=\mathrm{\mid }{\mathrm{cl}}_{\mathbb{Q}}(G)\mathrm{\mid }$. 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.