On commutative tensor factors of group algebras

TL;DR

This paper proves that commutative tensor factors of modular group algebras correspond to direct product decompositions of the underlying group, extending prior results and answering specific open questions.

Contribution

It establishes a link between tensor factorizations with commutative factors and group structure decompositions in modular group algebras.

Findings

Tensor product factors with commutative components correspond to direct group decompositions.

Extends previous work by Carlson and Kovács on the commutative case.

Answers specific open questions in the theory of modular group algebras.

Abstract

We prove that any tensor product factorization with a commutative factor of a modular group algebra over a prime field comes from a direct product decomposition of the group basis. This extends previous work by Carlson and Kov\'acs for the commutative case and answers a question of them in some cases.