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On the group pseudo-algebra of finite groups

Mark L. Lewis, Quanfu Yan

Abstract

Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in{\rm Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree $d\in {\rm Cod}(G)$. We show that there exist two finite $p$-groups with distinct orders that have the same group pseudo-algebra, providing an answer to Question 3.2 in \cite{Moreto2023}. In addition, we also discuss under what hypothesis two $p$-groups with the same group pseudo-algebra will be isomorphic.

On the group pseudo-algebra of finite groups

Abstract

Let be a finite group. The group pseudo-algebra of is defined as the multi-set where is the number of irreducible characters of with codegree . We show that there exist two finite -groups with distinct orders that have the same group pseudo-algebra, providing an answer to Question 3.2 in \cite{Moreto2023}. In addition, we also discuss under what hypothesis two -groups with the same group pseudo-algebra will be isomorphic.
Paper Structure (2 sections, 6 theorems, 3 equations)

This paper contains 2 sections, 6 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1.1

Let $p$ be a prime. There exists an abelian $p$-group $A$ and a group $G$ with $C(G)=C(A)$ so that $G$ is not isomorphic to $A.$ Hence, groups may have different orders even though they have the same group pseudo-algebra.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3