The nonvanishing first Hochschild cohomology of twisted finite simple group algebras

TL;DR

This paper proves that for any twisted group algebra of a finite simple group over an algebraically closed field of prime characteristic, the first Hochschild cohomology group is always nonzero, revealing persistent cohomological properties.

Contribution

It establishes that the first Hochschild cohomology of twisted group algebras of finite simple groups never vanishes, regardless of the 2-cocycle used.

Findings

First Hochschild cohomology is nonzero for all twisted group algebras of finite simple groups.

The result holds for any 2-cocycle in the second cohomology group.

This reveals a fundamental cohomological property of twisted group algebras in prime characteristic.

Abstract

Let $G$ be a finite simple group and $k$ be an algebraically closed field of prime characteristic dividing the order of $G$. We show that for all $2$-cocycles $\alpha \in Z^2(G;k^\times)$, the first Hochschild cohomology group of the twisted group algebra $HH^1(k_\alpha G)$ is nonzero.