Block Form of Frobenius Groups

TL;DR

This paper explores the structure of Frobenius groups through block theory, establishing new relationships between group algebra blocks, conjugate classes, and nilpotent properties.

Contribution

It introduces a block form of Brauer permutation Lemma and defines Frobenius corresponding blocks, linking group structure and nilpotency in the context of Frobenius groups.

Findings

Established a block form of Brauer permutation Lemma.

Defined Frobenius corresponding blocks between a group and its normal subgroup.

Proved connections between nilpotent properties and Frobenius blocks.

Abstract

The aim of this paper is to apply character properties of Frobenius group to a local block form of an group algebra. We start by establishing a block form of Brauer permutation Lemma by using block participation of conjugate classes of a group $G$. Then we can define a pair of Frobenius corresponding blocks between a group $G$ and its normal subgroup $N$. A near group condition is given to determine a pair of Frobenius corresponding blocks. With a pair of Frobenius corresponding blocks, we study its group structure. At last we prove connections between nilpotent properties and Frobenius corresponding blocks.