Loewy lengths of centers of blocks and exponents of defect groups

Abstract

In this paper we study the Loewy structure of the center $ZB$ of a block of a finite group with respect to an algebraically closed field of prime characteristic. We first state a new method for calculating the Loewy length $LL(ZB)$ of $ZB$ in terms of subsections and lower defect groups. By applying this result we give an upper bound of $LL(ZB)$ which depends on the exponent of a defect group of $B$ for some cases.