The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups

Abstract

Abstract Let 𝐺 be one of the sporadic simple Mathieu groups M 11 M_{11} , M 12 M_{12} , M 22 M_{22} , M 23 M_{23} or M 24 M_{24} , and suppose 𝑘 is 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 k ⁢ G kG . In particular, we calculate the dimension of HH 1 ⁢ ( B ) \mathrm{HH}^{1}(B) for the 𝑝-blocks 𝐵 of k ⁢ G kG , and in almost all cases, we determine whether HH 1 ⁢ ( B ) \mathrm{HH}^{1}(B) is a solvable Lie algebra.