Loewy lengths of centers of blocks II

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.