Group algebras in which the socle of the center is an ideal

TL;DR

This paper investigates the structure of finite groups over a field of characteristic p, focusing on when the socle of the center of their group algebra forms an ideal, and classifies such p-groups.

Contribution

It provides a classification of finite p-groups where the socle of the center of the group algebra is an ideal, and describes groups with the Reynolds ideal as an ideal.

Findings

Classified finite p-groups with socle of center as an ideal

Explicit description of groups with Reynolds ideal as an ideal

Structural insights into group algebras in characteristic p

Abstract

Let $F$ be a field of characteristic $p > 0$. We study the structure of the finite groups $G$ for which the socle of the center of $FG$ is an ideal in $FG$ and classify the finite $p$-groups $G$ with this property. Moreover, we give an explicit description of the finite groups $G$ for which the Reynolds ideal of $FG$ is an ideal in $FG$.