On centers of blocks with non-cyclic defect groups

TL;DR

This paper investigates the structure of the center of blocks with non-cyclic defect groups in finite groups, providing an improved upper bound on the Loewy length of the center over fields of prime characteristic.

Contribution

It improves Okuyama's upper bound on the Loewy length of the center for blocks with non-cyclic defect groups.

Findings

Established a tighter upper bound for the Loewy length of the center

Focused on blocks with non-cyclic defect groups

Enhanced understanding of block centers in modular representation theory

Abstract

In this short note we study the center ZB of a block B of a finite group over an algebraically closed field of prime characteristic through its Loewy length llZB. A result of Okuyama in 1981 gave an upper bound for llZB in terms of defect group of B. The purpose of this note is to improve this bound for non-cyclic defect groups.