Non-vanishing elements in finite groups

TL;DR

This paper proves that in finite groups with a normal nilpotent subgroup, certain elements in the intersection of the subgroup and the center of a Sylow p-subgroup do not vanish on any irreducible character, extending understanding of character evaluations.

Contribution

It establishes a new non-vanishing result for irreducible characters in finite groups with specific subgroup structures, linking normal nilpotent subgroups and Sylow p-subgroups.

Findings

Irreducible characters do not vanish on elements in N∩Z(P) under given conditions

Extends previous results on character non-vanishing in finite groups

Provides a new criterion for non-vanishing based on subgroup intersections

Abstract

Many results have been established about determining whether or not an element evaluates to zero on an irreducible character of a group. In this note it is shown that if a group $G$ has a normal nilpotent subgroup $N$, and $P$ is a Sylow $p$-subgroup of $G$, then no irreducible character of $G$ vanishes on $N\cap Z(P)$.