Inner and Outer Derivations of $\mathbb{F}V_{8n}$

TL;DR

This paper classifies all derivations of the group algebra of a specific group, detailing when they are inner or outer, depending on the field's characteristic and its relation to the group's order.

Contribution

It provides an explicit classification of all derivations of the group algebra of a complex group, including bases and dimensions, and characterizes when derivations are inner or outer.

Findings

All derivations are inner when the characteristic is 0 or relatively prime to n.

Outer derivations exist only when the characteristic divides n.

The derivation algebra's basis and dimension are explicitly given.

Abstract

Let $\mathbb{F}$ be a field of characteristic $0$ or an odd rational prime $p$. In this article, we give an explicit classification of all the inner and outer derivations of the group algebra $\mathbb{F}V_{8n}$, where $V_{8n}$ is a group of order $8n$ ($n$ a positive integer) with presentation $\langle a, b \mid a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. First, we explicitly classify all the $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$ by giving the dimension and a basis of the derivation algebra consisting of all $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$. Consequently, we classify all inner and outer derivations of $\mathbb{F}V_{8n}$ when $\mathbb{F}$ is an algebraic extension of a prime field. Thus, we establish that all the derivations of $\mathbb{F}V_{8n}$ are inner when the characteristic of $\mathbb{F}$ is $0$ or $p$ with $p$ relatively prime to $n$, and that non-zero outer derivations exist only in the case when the characteristic of $\mathbb{F}$ is $p$ with $p$ dividing $n$.