On the group pseudo-algebra of finite groups

Abstract

Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in \mathrm{Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree $d\in \mathrm{Cod}(G)$. 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.