Determination of Conjugacy Class Sizes from Products of Characters

TL;DR

This paper establishes new methods to determine conjugacy class sizes and the existence of classes with p-defect 0 by analyzing products of characters, extending previous character-based theorems.

Contribution

It introduces analogs to Robinson's and Strunkov's theorems by switching from character degrees to conjugacy class sizes and counts of trivial character multiplicities.

Findings

Conjugacy class sizes can be determined from products of characters.

Existence of p-defect 0 conjugacy classes can be inferred from character products.

New character-based criteria for conjugacy class properties are established.

Abstract

Robinson showed that the character degrees are determined by knowing, for all $n$, the number of ways that the identity can be expressed as a product of $n$ commutators. Earlier, Strunkov showed that the existence of characters of $p$-defect 0 can be determined by counting solutions to certain equations involving commutators and conjugates. In this paper, we prove analogs to Robinson's and Strunkov's theorems by switching conjugacy classes and characters. We show that counting the multiplicity of the trivial character in certain products of characters determines the conjugacy class sizes and existence of conjugacy classes with $p$-defect 0.