Notes on Loewy series of centers of $p$-blocks

TL;DR

This paper investigates the Loewy series of centers of p-blocks in modular group algebras, providing results that relate the structure of defect groups to the number of irreducible characters.

Contribution

It establishes a lower bound of p+2 on the number of characters for p-blocks with defect groups containing non-elementary centers.

Findings

A p-block with a defect group containing a non-elementary center has at least p+2 irreducible characters.

Results connect the Loewy series structure to character counts in modular representation theory.

Provides new insights into the structure of centers of p-blocks in finite groups.

Abstract

The present paper describes some results on the Loewy series of the center of a modular group algebra in order to solve a problem on the number of irreducible ordinary characters. For instance, we prove that a $p$-block of a finite group has at least $p+2$ characters if its defect group contains non-elementary center.