On k(D)-blocks

TL;DR

This paper investigates the relationship between blocks of finite groups and their defect groups, introducing the concept of strongly $k(D)$-blocks and characterizing cyclic defect groups via inertial index.

Contribution

It introduces the notion of strongly $k(D)$-blocks and provides a necessary and sufficient condition for blocks with cyclic defect groups based on inertial index.

Findings

Defined strongly $k(D)$-blocks.

Characterized cyclic defect groups using inertial index.

Established criteria for $k(D)$-blocks in finite groups.

Abstract

The objective of this research paper is to study the relationship between a block of a finite group and a defect group of such block. We define a new notion which is called a strongly $k(D)$-block and give a necessary and sufficient condition of a block with a cyclic defect group to be a $k(D)$-block in term of its inertial index.