Non-noetherian groups and primitivity of their group algebras

TL;DR

This paper establishes conditions under which the group algebra of a group is primitive, particularly for groups with certain free subgroups and a specific combinatorial property, extending previous results and applying to one relator groups with torsion.

Contribution

It introduces a new condition $(igstar)$ that guarantees the primitivity of group algebras for a broad class of groups, generalizing earlier work and covering new cases.

Findings

Proves group algebra primitivity under condition $(igstar)$.

Shows that countably infinite groups satisfying $(igstar)$ have primitive group algebras.

Determines primitivity for group algebras of one relator groups with torsion.

Abstract

We prove that the group algebra $KG$ of a group $G$ over a field $K$ is primitive, provided that $G$ has a free subgroup with the same cardinality as $G$, and that $G$ satisfies the following condition $(\ast)$: for each subset $M$ of $G$ consisting of a finite number of elements not equal to $1$, and for any positive integer $m$, there exist distinct $a$, $b$, and $c$ in $G$ so that if $(x_{1}^{-1}g_1x_{1}) \cdots (x_{m}^{-1}g_mx_{m})=1$, where $g_i$ is in $M$ and $x_i$ is equal to $a$, $b$, or $c$ for all $i$ between $1$ and $m$, then $x_{i}=x_{i+1}$ for some $i$. This generalizes results of \cite{Bal}, \cite{For}, \cite{Ni07}, and \cite{Ni11}, and proves that, for every countably infinite group $G$ satisfying $(\ast)$, $KG$ is primitive for any field $K$. We use this result to determine the primitivity of group algebras of one relator groups with torsion.