On solvability of the first Hochschild cohomology of odd p-groups

TL;DR

This paper establishes a precise criterion for when the first Hochschild cohomology Lie algebra of an odd p-group is solvable, using a graph-based approach to analyze its structure.

Contribution

It introduces a new graph-based criterion for the solvability of HH^1(kP) Lie algebras of odd p-groups, advancing understanding of their algebraic structure.

Findings

Provides necessary and sufficient conditions for solvability

Connects group structure to graph properties

Offers new insights into the Lie algebra structure of HH^1(kP)

Abstract

We give a necessary and sufficient criterion for the solvability of $\operatorname{HH}^1(kP)$ as a Lie algebra, where $P$ is a $p$-group with $p$ odd, in terms of a directed graph constructed from the group $P$. This gives non-trivial results on the structure of such Lie algebras.