Outer derivations on blocks of group algebras

Abstract

Let $G$ be a finite group whose order is divisible by the characteristic of a field $k$. If $B$ is a block of $kG$ with defect group $P$, we prove that the space of derivations on $kP$ which are restrictions of derivations on $kG$, modulo inner derivations, is isomorphic to a subspace of $\operatorname{HH}^1(B,B)$. Using this, we provide various group theoretic criteria for the non-vanishing of $\operatorname{HH}^1(B,B)$. In particular, we show $\operatorname{HH}^1(B,B)\neq 0$ for principal blocks having abelian defect group, for all blocks of the symmetric and alternating groups, for blocks of finite groups of Lie type in defining characteristic, and for blocks of general linear groups in any characteristic. Building on this, we show that if $k$ has prime characteristic $p>5$, and if $B$ is any block of $kG$ with Sylow defect group, then $\operatorname{HH}^1(B,B)\neq 0$. By the same method we also prove that if $k$ has prime characteristic $p>5$, then the first Hochschild cohomology group of any twisted group algebra is non-zero.