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 $|D|/p+p-1$ provided $D$ is not cyclic. If $D$ is nonabelian, 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 $LL(ZB)\geqslant \min \{p^{d-1},4p^{d-2}\}$. This extends some results previously obtained by the present authors. Moreover, we characterize blocks with uniserial center.