The Twisted Derivation Problem for Group Rings

TL;DR

This paper investigates twisted derivations in group rings with specific structural conditions, generalizes previous results, and explores their implications for important conjectures and problems in algebra.

Contribution

It extends the understanding of $(\sigma, au)$-derivations in group rings, providing new conditions for their triviality and connecting them to major algebraic conjectures.

Findings

Existence of a ring extension where certain cohomology groups vanish

Application of results to integral group rings of finite groups

Construction of examples with non-inner twisted derivations

Abstract

We study $(\sigma,\tau)$-derivations of a group ring $RG$ where $G$ is a group with center having finite index in $G$ and $R$ is a semiprime ring with $1$ such that either $R$ has no torsion elements or that if $R$ has $p$-torsion elements, then $p$ does not divide the order of $G$ and let $\sigma,\tau$ be $R$-linear endomorphisms of $RG$ fixing the center of $RG$ pointwise. We generalize Main Theorem $1.1$ of \cite{Chau-19} and prove that there is a ring $T\supset R$ such that $\mathcal{Z}(T)\supset\mathcal{Z}(R)$ and that for the natural extensions of $\sigma, \tau$ to $TG$ we get $H^1(TG,{}_\sigma TG_\tau)=0$, where ${}_\sigma TG_\tau$ is the twisted $TG-TG$-bimodule. We provide applications of the above result and Main Theorem $1.1$ of \cite{Chau-19} to integral group rings of finite groups and connect twisted derivations of integral group rings to other important problems in the field such as the Isomorphism Problem and the Zassenhaus Conjectures. We also give an example of a group $G$ which is both locally finite and nilpotent and such that for every field $F$, there exists an $F$-linear $\sigma$-derivation of $FG$ which is not $\sigma$-inner.