More on regular subgroups of the affine group

This paper is a new contribution to the study of regular subgroups of the affine group AGLn(F)AGLn(F), for any field FF. In particular we associate to each partition λ≠(1n+1)λ≠(1n+1) of n+1n+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≤4n≤4. Our classification is equivalent to the classification of split local algebras of dimension n+1n+1 over FF. Our methods, based on classical results of linear algebra, are computer free.