Characters, Hall subgroups, and Normal Complements

TL;DR

This paper extends a classical result about the extension of irreducible characters from Hall subgroups to the whole group, providing a modular version under broader conditions.

Contribution

It generalizes a 1989 theorem by establishing a modular analogue with more general hypotheses for character extension in finite groups.

Findings

Provides a modular version of the character extension theorem

Broadens the conditions under which characters extend to the entire group

Connects Hall subgroups and normal complements in a new context

Abstract

If $H$ is a Hall subgroup of a finite group $G$, it was proven in 1989 using the classification of finite simple groups that all the irreducible complex characters of $H$ extend to $G$ if and only if there is $N\trianglelefteq G$ such that $HN=G$ and $H\cap N=1$. In this note we offer a modular version of this result under more general hypothesis.