Loewy lengths of centers of blocks II

TL;DR

This paper establishes bounds on the Loewy length of the center of blocks in finite groups with non-cyclic defect groups, classifies blocks with large Loewy length, and characterizes blocks with uniserial centers.

Contribution

It provides new bounds for Loewy lengths of centers of blocks with non-cyclic defect groups and classifies blocks with high Loewy length, extending previous results.

Findings

Bound LL(ZB) by |D|/p + p - 1 for non-cyclic D.

Stronger bound LL(ZB) < min{p^{d-1}, 4p^{d-2}} for non-abelian D.

Classification of blocks with LL(ZB) ≥ min{p^{d-1}, 4p^{d-2}}.

Abstract

Let ZB be the center of a p-block B of a finite group with defect group D. We show that the Loewy length LL(ZB) of ZB is bounded by $\frac{|D|}{p}+p-1$ provided D is not cyclic. If D is non-abelian, we prove the stronger bound $LL(ZB)<\min\{p^{d-1},4p^{d-2}\}$ where $|D|=p^d$. Conversely, we classify the blocks B with $LL(ZB)\ge\min\{p^{d-1},4p^{d-2}\}$. This extends some results previously obtained by the present authors. Moreover, we characterize blocks with uniserial center.