Converse of Proposition 2.19 for τ-tilting finiteness

Prove the converse of Proposition 2.19 for arbitrary finite groups G: given the p-hyperfocal subgroup R of G, show that if the group algebra kG (over an algebraically closed field of characteristic p) is τ-tilting finite, then R must be cyclic, or (when p = 2) R must be dihedral, semidihedral, or generalized quaternion. Equivalently, characterize τ-tilting finiteness of kG solely in terms of the p-hyperfocal subgroup R.

Background

The paper studies how subgroup invariants of a finite group G control τ-tilting finiteness of the group algebra kG. The p-hyperfocal subgroup R (the intersection of a Sylow p-subgroup with Op(G)) is proposed as the controlling invariant.

Proposition 2.19 provides sufficient conditions: if R is cyclic, or if p=2 and R is dihedral, semidihedral, or generalized quaternion, then kG is τ-tilting finite. The conjecture asks for the converse, giving a complete classification in terms of R.

The present work supplies additional evidence by proving the conjectural direction for generalized symmetric groups (Z/mZ)n ⋊ H under suitable hypotheses (Theorem 4.11), extending earlier confirmations in the special case G = P ⋊ H with P and H abelian.

References

In [HK], we conjectured that the converse of Proposition 2.19 also holds, and verified that it is true in the case G = P ⋊ H, where P is an abelian p-group and H is an abelian p′-group acting on P.

— $τ$-Tilting finiteness of group algebras over generalized symmetric groups  (2405.10726 - Hiramae, 2024) in Section 2.2, after Proposition 2.19