On groups with at most five irrational conjugacy classes
Gabriel A. L. Souza
TL;DR
The paper investigates finite groups with few irrational conjugacy classes, establishing that if $G$ has at most five such classes, then the number of rational conjugacy classes equals the number of rational irreducible characters, $|\mathrm{cl}_{\mathbb{Q}}(G)| = |\mathrm{Irr}_{\mathbb{Q}}(G)|$, without invoking the Classification of Finite Simple Groups. The authors build a Galois-action framework on $\mathrm{Irr}(G)$ and $\mathrm{cl}(G)$, derive a detailed analysis for the case of four irrational classes, and show that the equality propagates to the other low-count cases ($2$, $3$, $5$) via BLCT-type results, with sharpness demonstrated by explicit counterexamples for extending beyond five. They then derive prime-divisor bounds for solvable groups with two or three irrational conjugacy classes using the $k$-eigenvalue property and Brauer characters, and prove a general bound showing the number of irrational irreducible characters is at most $n^2/2$ when there are $n$ irrational conjugacy classes (and vice versa). The results reveal a duality with known rational-case questions and contribute to understanding how fields generated by character values constrain group structure, all while remaining independent of CFSG.
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 $χ\in \operatorname{Irr}(G)$ such that $χ(K) \not \in \mathbb{Q}$. 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.