The Lie algebra structure of the $HH^1$ of the blocks of the sporadic Mathieu groups

Abstract

Let $G$ be a sporadic Mathieu group and $k$ an algebraically closed field of prime characteristic $p$, dividing the order of $G$. In this paper we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the $p$-blocks of $kG$. In particular, letting $B$ denote a $p$-block of $kG$, we calculate the dimension of $HH^1(B)$ and in the majority of cases we determine whether $HH^1(B)$ is a solvable Lie algebra.