Above Isaacs' Head Characters

TL;DR

This paper extends the understanding of character theory in finite groups by establishing a local determination result for Isaacs' head characters of normal solvable subgroups, and provides a new lower bound on conjugacy classes.

Contribution

It introduces a new result for Isaacs' head characters of normal solvable subgroups and derives a lower bound on the number of conjugacy classes based on Carter subgroups.

Findings

Character theory above p'-degree characters is locally determined.

A new lower bound for conjugacy classes in finite groups.

Extension of inductive McKay condition implications.

Abstract

The proof of the inductive McKay condition has been shown to imply that the character theory above the characters of degree not divisible by $p$ of a normal subgroup is locally determined. In this note, we establish a similar result for the Isaacs' head characters of a normal solvable subgroup of an arbitrary group. In particular, we give a new lower bound of the number of conjugacy classes of a finite group in terms of the Carter subgroups of any of its normal solvable subgroups.