On a question by Roggenkamp about group algebras

TL;DR

This paper explores conditions under which the group algebra of a finite group over localized integers is semiperfect, providing a complete arithmetic criterion in the ordinary case and proposing a conjecture for the modular case.

Contribution

It offers a necessary and sufficient arithmetic criterion for semiperfectness in the ordinary case and extends the discussion with a conjecture for the modular case.

Findings

Arithmetic criterion for semiperfectness in the ordinary case

Proposal of a conjecture extending the criterion to the modular case

Advances understanding of group algebra structures over localized integers

Abstract

We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a conjecture, which extends the criterion.