On the group pseudo-algebra of finite groups (Q6644097)

The paper is about the concept of the group pseudo-algebra for finite groups, which was first introduced by \textit{A. Moretó} [Bull. Lond. Math. Soc. 55, No. 1, 234--241 (2023; Zbl 1525.20007)] and the author investigates here its properties and implications. Recall that the \textit{group pseudo-algebra} of a finite group \( G \) is a multiset\N\[\NC(G) = \{(d, m_G(d)) \mid d \in \mathrm{Cod}(G)\},\N\]\Nwhere \( \mathrm{Cod}(G) \) is the set of codegrees of the irreducible characters of \( G \) and \( m_G(d) \) counts how many irreducible characters have codegree \( d \). A codegree is calculated as\N\[\N\operatorname{cod}_\chi = \frac{|G : \ker(\chi)|}{\chi(1)},\N\]\Nwhere \( \chi \) is an irreducible character, \( \ker(\chi) \) is its kernel, and \( \chi(1) \) is its degree.\N\NOne of the main theorems shows that two finite \( p \)-groups can have the same group pseudo-algebra even if their orders differ, thereby answering an open question in the field. Another result focuses on the conditions under which groups with the same group pseudo-algebra are isomorphic. For instance, if \( G \) is metacyclic or if its derived subgroup is generated by two elements, then \( G \) is isomorphic to an abelian group \( A \) that has the same group pseudo-algebra. The paper is enriched with many results and nice examples.