More on regular subgroups of the affine group

TL;DR

This paper classifies abelian regular subgroups of the affine group over any field, linking them to partitions of n+1 and split local algebras, providing a non-computer-based classification for small dimensions.

Contribution

It introduces a partition-based classification of abelian regular subgroups of the affine group and classifies certain types for dimensions up to 4 without computational methods.

Findings

Different partitions define non-conjugate subgroups.

Classified regular subgroups for n ≤ 4.

Linked subgroup classification to split local algebras.

Abstract

This paper is a new contribution to the study of regular subgroups of the affine group $AGL_n(F)$, for any field $F$. In particular we associate to any partition $\lambda\neq (1^{n+1})$ of $n+1$ abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for $n\leq 4$. Our classification is equivalent to the classification of split local algebras of dimension $n+1$ over $F$. Our methods, based on classical results of linear algebra, are computer free.