Supports of irreducible characters of p-groups

TL;DR

This paper investigates the supports of irreducible characters in specific p-groups, providing bounds on the size and element orders in the support for classes like metabelian and nilpotent groups.

Contribution

It establishes new bounds on the supports of irreducible characters for certain p-groups, including metabelian and nilpotent groups, with explicit size and order constraints.

Findings

Support of irreducible characters in metabelian p-groups contains at most |P|/chi(1)^2 conjugacy classes.

Bounds on element orders in the support of characters are provided.

More precise results are given for p-groups of nilpotence class at most 3 and order dividing p^9.

Abstract

If chi is an irreducible character of a finite group G then the support of chi is the subset of G on which chi does not vanish. In this note, we study the supports of characters of certain classes of p-groups (a p-group is a finite group of prime power order). We show that if chi is an irreducible character of a metabelian p-group P, then the support of chi contains at most |P|/chi(1)^2 conjugacy classes, and we give a bound on the orders of the elements in the support. We give more precise results for p-groups of nilpotence class at most 3 and groups of order dividing p^9.