Nonvanishing of cohomology for groups

Tiago Cruz; Rene Marczinzik

paper

Nonvanishing of cohomology for groups

arXiv:2609.24573

Abstract

Let $G$ be a finite group. We prove that the order of the abelianization $G / G^{'}$ divides the order of $H^{2 m} (G, Z)$ for every $m \geq 1$ . As a consequence, for a field $K$ , nonvanishing of $E x t_{K G}^{1} (K, K)$ implies nonvanishing of $E x t_{K G}^{n} (K, K)$ for every $n \geq 1$ . This answers a conjecture by Erdmann, Klász and Marczinzik. We use this to show that the Hochschild cohomology of $K G$ is nonzero in every nonnegative degree whenever the characteristic of $K$ divides $∣ G ∣$ .