$\tau$-Tilting finiteness of group algebras of semidirect products of abelian $p$-groups and abelian $p'$-groups

TL;DR

This paper investigates the conditions under which group algebras of certain finite groups are $ au$-tilting finite, showing that $p$-hyperfocal subgroups play a key role in determining this property for groups formed as semidirect products of abelian $p$-groups and abelian $p'$-groups.

Contribution

It demonstrates that $ au$-tilting finiteness of group algebras for semidirect products of abelian $p$-groups and abelian $p'$-groups is controlled by their $p$-hyperfocal subgroups, providing new insights into the structure of these algebras.

Findings

$ au$-tilting finiteness is determined by $p$-hyperfocal subgroups.

Tame blocks of group algebras are always $ au$-tilting finite.

The structure of the group algebra influences $ au$-tilting properties through subgroup analysis.

Abstract

Demonet, Iyama and Jasso introduced a new class of finite dimensional algebras, $\tau$-tilting finite algebras. It was shown by Eisele, Janssens and Raedschelders that tame blocks of group algebras of finite groups are always $\tau$-tilting finite. Given the classical result that the representation type (representation finite, tame or wild) of blocks is determined by their defect groups, it is natural to ask what kinds of subgroups control $\tau$-tilting finiteness of group algebras or their blocks. In this paper, as a positive answer to this question, we demonstrate that $\tau$-tilting finiteness of a group algebra of a finite group $G$ is controlled by a $p$-hyperfocal subgroup of $G$ under some assumptions on $G$. We consider a group algebra of a finite group $P\rtimes H$ over an algebraically closed field of positive characteristic $p$, where $P$ is an abelian $p$-group and $H$ is an abelian $p'$-group acting on $P$, and show that $p$-hyperfocal subgroups determine $\tau$-tilting finiteness of the group algebras in this case.