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

TL;DR

This paper investigates the Lie algebra structure of the first Hochschild cohomology groups of blocks of Mathieu groups over fields of prime characteristic, providing dimension calculations and solvability criteria.

Contribution

It offers the first detailed description of the Lie algebra structure of $HH^1$ for blocks of Mathieu groups in prime characteristic.

Findings

Calculated the dimension of $HH^1(B)$ for various blocks.

Determined solvability of $HH^1(B)$ in most cases.

Provided new insights into the algebraic structure of Mathieu group blocks.

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.