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A note on the codegree of finite groups

Mark L. Lewis, Quanfu Yan

Abstract

Let $χ$ be an irreducible character of a group $G,$ and $S_c(G)=\sum_{χ\in {\rm Irr}(G)}{\rm cod}(χ)$ be the sum of the codegrees of the irreducible characters of $G.$ Write ${\rm fcod} (G)=\frac{S_c(G)}{|G|}.$ We aim to explore the structure of finite groups in terms of ${\rm fcod} (G).$ On the other hand, we determine the lower bound of $S_c(G)$ for nonsolvable groups and prove that if $G$ is nonsolvable, then $S_c(G)\geq S_c(A_5)=68,$ with equality if and only if $G\cong A_5.$ Additionally, we show that there is a solvable group so that it has the codegree sum as $A_5.$

A note on the codegree of finite groups

Abstract

Let be an irreducible character of a group and be the sum of the codegrees of the irreducible characters of Write We aim to explore the structure of finite groups in terms of On the other hand, we determine the lower bound of for nonsolvable groups and prove that if is nonsolvable, then with equality if and only if Additionally, we show that there is a solvable group so that it has the codegree sum as
Paper Structure (3 sections, 6 theorems, 5 equations)

This paper contains 3 sections, 6 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. The invariant ${\rm fcod}(G)$
  3. Lower bound of $S_c(G)$ for nonsolvable groups

Key Result

Theorem 1.1

Let $G$ be a non-solvable group. Then $S_c(G)\geqslant S_c(A_5)=68,$ with equality if and only if $G\cong A_5.$

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • Example 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1