Nilpotent Decomposition in Integral Group Rings

TL;DR

This paper investigates the nilpotent decomposition property in integral group rings, classifies finite groups with this property, and characterizes those with a single non-division Wedderburn component.

Contribution

It provides a classification of finite SSN groups with a unique non-division Wedderburn component in their rational group algebra.

Findings

Identified conditions under which groups have the nilpotent decomposition property.

Classified finite SSN groups with a single non-division Wedderburn component.

Abstract

A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $\alpha$ of the integral group ring $\mathbb{Z}[G]$ one has that $\alpha e$ also belong to $\mathbb{Z}[G]$, for every primitive central idempotent $e$ of the rational group algebra $\mathbb{Q}[G]$. Results of Hales, Passi and Wilson, Liu and Passman show that this property is fundamental in the investigations of the multiplicative Jordan decomposition of integral group rings. If $G$ and all its subgroups have ND then Liu and Passman showed that $G$ has property SSN, that is, for subgroups $H$, $Y$ and $N$ of $G$, if $N\lhd H $ and $Y\subseteq H$ then $N\subseteq Y$ or $YN$ is normal in $H$; and such groups have been described. In this article, we study the nilpotent decomposition property in integral group rings and we classify finite SSN groups $G$ such that the rational group algebra $\mathbb{Q}[G]$ has only one Wedderburn component which is not a division ring.