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On the sum of character codegrees of finite groups

Mark L. Lewis, Quanfu Yan

Abstract

Let $χ$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{χ\in {\rm Irr}(G)}{\rm cod}(χ).$ We consider the question if $S_c(G)\leq S_c(C_n)$ is true for any finite group $G,$ where $n=|G|$ and $C_n$ is a cyclic group of order $n.$ We show this inequality holds for many classes of groups. In particular, we provide an affirmative answer for any finite group whose order is divisible by up to 99 primes. However, we show that the question does not hold true in all cases, by evidence of a counterexample.

On the sum of character codegrees of finite groups

Abstract

Let be an irreducible character of a group We denote the sum of the codegrees of the irreducible characters of by We consider the question if is true for any finite group where and is a cyclic group of order We show this inequality holds for many classes of groups. In particular, we provide an affirmative answer for any finite group whose order is divisible by up to 99 primes. However, we show that the question does not hold true in all cases, by evidence of a counterexample.
Paper Structure (4 sections, 13 theorems, 27 equations)

This paper contains 4 sections, 13 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groups that satisfy $S_c(G)\leqslant S_{c}(C_{n})$
  4. A family of examples to Question A.

Key Result

Theorem 1.1

If $G$ is a nilpotent group of order $n$, then $S_{c}(G)\leqslant S_{c}(C_{n})$ with equality if and only if $G$ is cyclic.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • ...and 3 more