# Fake Galois Actions

## TL;DR

This paper establishes the existence of fake Galois actions on irreducible Brauer characters for universal covers of non-abelian simple groups, aiding the proof of an $ ext{l}$-modular Glauberman-Isaacs correspondence.

## Contribution

It proves the existence of fake Galois actions for all non-abelian simple groups' universal covers, a key step in modular representation theory.

## Key findings

- Existence of fake Galois actions for all non-abelian simple groups' universal covers.
- Provides an inductive condition for $ ext{l}$-modular Glauberman-Isaacs correspondence.
- Advances understanding of modular representation theory of finite groups.

## Abstract

We prove that for all non-abelian finite simple groups $S$, there exists a fake mth Galois action on IBr$(X)$ with respect to $X \lhd X \rtimes $ Aut$(X)$, where $X$ is the universal covering group of $S$ and $m$ is any non-negative integer coprime to the order of $X$. This is one of the two inductive conditions needed to prove an $\ell$-modular analogue of the Glauberman-Isaacs correspondence.