Finite simple groups with short Galois orbits on conjugacy classes

TL;DR

This paper classifies finite simple groups based on the size of Galois orbits on conjugacy classes, identifying those with orbits of size at most 4, and explores their associated algebraic structures.

Contribution

It provides a complete classification of finite simple groups with small Galois orbits on conjugacy classes and characterizes their integral group ring units.

Findings

Finite simple groups with Galois orbits of size ≤ 4 are classified.

Identifies groups where the normalized group of central units of ZG is infinite cyclic.

Establishes a link between Galois orbit sizes and algebraic properties of group rings.

Abstract

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.